It has been entirely too long since I have used ArcGIS. So, I made a map.
My methodology for the map is as follows. I took as my base a map of the world's elevation, both above and below sea level. Using ArcMap, I then simply flipped the coloration--oceans are shown in green and land is shown in blue. For the purposes of this fictional world, then, any location in the real world with an elevation of 1000 meters will have an elevation of -1000 meters. I then committed one of the cardinal sins of cartography--I did not include a legend. In the interest of getting this published in a timely manner, I won't go back and make one; I plead the cartography gods forgiveness for my omission. In the oceans, the darker the blue, the deeper the ocean. On the land, the lighter or yellower the green, the higher the elevation. Each change in shade represents a change of 1000 meters.
I have set up the picture so that it changes in size with your browser's window size. If it looks grainy, make the window smaller. If you're looking at this on your phone... basically, I am sorry.
Now, there are a couple important things to remember. First, this map projection (like any other map projection) involves a great deal of distortion. In this case, the map preserves direction but greatly distorts size as you get further from the equator. Basically, trust the Ginger's numbers rather than my picture when you're thinking about proportions.
Now, the geography of this world.
One of the first things you'll notice is that, with the exception of a very small number of islands (lakes whose base extends below sea level), the entire world's landmass is connected in a single unit. The oceans are divided into three main groups, which I'll call the Eastern Ocean (Europe, Asia, and Africa), the Western Ocean (North and South America), and the Southern Ocean (Antarctica). The next-largest body of water is the Great Australian Lake.
The coastlines pose an interesting situation. Although there are extensive lowlands on the northern borders of the Eastern and Western Oceans, in most cases the elevation drops very sharply from the World Continent to the oceans. This is the result of the swift dropoffs at the continental shelves in our real world. Even in the low-lying areas, often there is a huge cliff followed by miles of gently sloping land.
Looking at the map, you can see another interesting change--there are very few mountains. Elevation changes very slowly, so although it does vary greatly, the vast majority of the World Continent is more or less flat. In the Atlantic Belt of the continent, there is a valley running down the middle of the belt. In our real world, that is where two plates of the Earth's crust are diverging and magma is rising to the ocean floor to fill the gap. The Indian Protrusion and the Pacific Belt have more complex topographies, but the basic pattern remains the same. The biggest exception is the Mariana Mountains, along the eastern coast of the Eastern Ocean. The mountains are narrow but exceptionally tall.
Islands are few and far between. The largest by far is the Caspian Island in the Eastern Ocean. Although other islands do exist, they are all tiny. (Remember, not all lakes become islands. The only islands are the result of lakes whose floor is below sea level.) The Mediterranean Peninsula is so narrowly connected to the World Continent that it is almost an island, but it is still connected via the Isthmus of Gibraltar.
Lakes, on the other hand, are a bit more common. The Great Australian Lake is the largest, of course, and you could even call it an ocean in the same sense that you'd call Pluto a planet. Other major lakes include the British Lakes, the Japanese Lakes, the Indonesian and Lesser Australian Lakes, Lake Madagascar, and Greenlake and Icelake. All of these are lakes that extend below sea level.
Now that we've established the basic topography of the world, let's examine how climactic and geological forces would affect it.
Our first big question is presented by the joint forces of erosion and gravity. Water is a much stronger force for erosion than air is, which is why I haven't bothered describing the topography of the new ocean floor--it will be smoothed very quickly. The rather smooth continents, on the other hand, have nothing to fear from erosion. The problems come where elevation changes abruptly. Depending on the underlying geology, the seaside cliffs and the Mariana Mountains could end up being smoothed over time. This is fiction, though, so you can make the geology be whatever you want it to be. Just beware of landslides.
Next, there is the climate. If we assume that it is only the land structure that has changed and the planet's orbit and tilt remain the same, then the Southern Ocean will be more or less permanently frozen over, much as the Arctic Ocean is in our world. The same is likely to be true of most or all of Greenlake. The northern lowlands of the World Continent will be much like the northernmost reaches of Siberia and Canada, on the outermost fringe of suitability for human habitation.
This may be modified, however, by a consideration I did not account for when I first started answering this question. While the original source of the Earth's heat energy is the Sun, that energy travels through outer space as electromagnetic radiation and is converted to heat after it reaches the Earth. While some of it is converted to heat in the upper atmosphere, about two-thirds of the electromagnetic radiation that is converted to heat is converted to heat at the planet's surface. Some of this heat remains on the surface, while some is transferred immediately back to the atmosphere. Not surprisingly, water and land behave very differently in this equation. In all honesty, I do not understand this process well enough to tell you how increased land surface area and decreased water surface area would affect this process. If I had to guess, though, I would say that the world would be slightly cooler. This has much more to do with physics than geography, though, and I am most definitely not an expert on physics.
One of the most important basic principles of meteorology is that land changes temperature more rapidly than water. The surface of a land area will both gain and lose heat relatively quickly. The surface and subsurface of a body of water will gain and lose it gradually, and will not reach the same extremes. There are reasons for this, but they're not terribly important as long as you understand the basic fact. Because of the interaction between surface temperature and air temperature, large bodies of water also exert a moderating effect on coastal regions. So, in general, temperatures (both hot and cold) will get more extreme the further you get inland.
The influence of ocean currents will be much less significant in our inverse world than it is in the real world. I do think they would probably exist, but they would be fewer, and they would not interact with each other. In the northern hemisphere, currents move in a clockwise direction; in the southern hemisphere, they move counterclockwise. My best guess is that there would be four of them--one in North America, one in South America, one in Eurasia, and one in Africa. In the real world, currents affect temperature by bringing warm equatorial water towards the poles and cold polar water towards the equator. The east coast of a landmass is generally heated by equatorial water, and the west coast is generally cooled by polar water. There are exceptions caused by non-circular currents, but in the inverse world the oceans would be too disconnected for anything like that to exist.
Certain contributors to temperature would not change at all, of course. For instance, higher elevations will generally be cooler than lower elevations. Areas closer to the poles will be colder, while areas closer to the equator will be warmer. Seasonal variations will exist. Their biggest effect will be in the areas between the equator and the poles, where summers will be hot and winters will be cold. Near the poles, it will be cold year-round; near the equator, it will be hot year-round.
Wind is a very complex topic and the differences probably won't be hugely important, so I won't spend much time on it. Suffice it to say that coastal areas and mountainous areas will have the most persistent winds. Generally, winds will move from sea to land during the day and during the summer, and from land to sea during the night and during the winter. The day-night distinction is more of a localized phenomenon, while the summer-winter distinction is more regional or global. It is this summer-winter reversal of air flow that causes monsoons in certain parts of the world, and you could expect to find a smaller but still significant monsoon effect in some parts of the inverse world. It would probably be most noticeable on the southern coast of the Eastern (Eurasian) ocean, although other equatorial regions near large bodies of water may experience it to one degree or another.
Now, precipitation. Precipitation is a complex process as well, but one constant element is that warm and wet air is forced upwards until it cools and the moisture condenses. This generally happens in one of three ways. Convective uplift (a vertical circular pattern of air movement) create large thunderclouds and is characteristic of warm parts of the world and warm seasons. Orographic lifting is caused by mountains; warm wet air goes up a mountain, cools, drops rain on the windward side of the mountain, and then descends the other side of the mountain both drier and hotter than it was when it started. (This is the cause of rain shadows, and the reason that the pattern of oceans followed by mountains followed by deserts is common.) Frontal lifting (and its less common variant, convergent lifting) is caused when two dissimilar bodies of air meet and the warmer body is forced above the colder body. This creates generalized steady precipitation and tends to occur in the middle latitudes where cold polar air meets hot tropical air. Bodies of air near the poles and the equator are too uniform for this type of precipitation to be common.
Globally, precipitation in our world is highest in the tropical regions. The trade winds move from east to west, so the east coasts of landmasses tend to be on the receiving end of higher precipitation. Coastal mountain ranges in northwestern North America and southwestern South America also produce high precipitation. I suspect that the tendency of the tropics to be particularly rainy would carry over to the inverse world. The west coast precipitation in the Americas obviously would not carry over. However, I suspect it might have a parallel on the eastern end of the Eurasian ocean, between the ocean and the Marianas Mountains. In all of these cases, rain shadows would be strongly accentuated.
Low precipitation in our world tends to occur on the western ends of continents in subtropical latitudes (i.e. 30 degrees), especially if there are mountains to the east. Inland areas may also be very dry simply because of the distance from large bodies of water. Also, the poles have essentially no precipitation. Although they may have permanent snow and ice cover, the complete absence of precipitation makes them technically deserts. All three of these effects would carry over to the inverse world. The inland effect, in all likelihood, would be by far the most significant. Much of the Pacific landmass would be extremely dry.
As a general rule, wet areas have little variation in yearly precipitation, and dry areas have a great deal of variation. Put another way, wet areas are always consistently wet, but dry areas are not always consistently dry.
Because of the placement of the oceans, hurricanes are extremely unlikely and the only region with any reasonable potential for hurricane development is in the northern portion of the African Ocean. (The reasons are too long to include; if you want details, Wikipedia is your friend.) These hurricanes would move northwest after their development, losing force after landfall just as they do in the real world.
As previously mentioned, there will be several large lakes. However, in this world, just as in our world, not all lakes will be below sea level. The lakes shown on the map will probably all be saltwater lakes. Any rain that falls, however, will be fresh water, and it will flow through rivers and freshwater lakes just as it does here and now. The inland valleys visible in the map will probably hold a large network of lakes; depending on the amount of rainfall, some of them may even become one long, narrow lake. Whether these lakes are freshwater or saltwater would depend on whether they are connected by rivers to the various oceans. Such a connection would usually be plausible. Rivers in the uplands will likely cut canyons leading either to the central lakes or to the oceans. Once they pass the cliffs and enter the lowlands, they will do what water always does at the base of a mountain: slow down and spread out.
All of this is very interesting, of course, but for the most part it's only background to the background of your story. The most important part of all this is how it affects where and how people live.
The first thing you can expect is a significant concentration of population in coastal areas. With the exception of the Antarctic Ocean, most coasts will contain major population centers. Population will also be higher near rivers and lakes. The Atlantic and Indian landmasses will contain such bodies of water, as will the western portions of the Pacific landmass. Conversely, dry inland areas will be sparsely populated. The Pacific landmass is the biggest example; most of it may very well be almost uninhabited. If it does have inhabitants, they will be nomadic rather than settled. In pre-modern eras, at least, access to water is one of the biggest prerequisites to the formation of (relatively) large cities, and severe lack of water may make nomadism the only realistic option for finding sufficient resources to raise food.
Beyond this, I think you'll be as good a judge of how to interpret the effects on life as I am. A good study of history, including where and how ancient cultures developed, will give you the tools necessary to make accurate representations of the inverse world's civilizations.
I'd apologize for the length of this answer, but really, I am not sorry at all. I have had entirely too much fun doing this, and I hope I get the excuse to do it again soon. In the interest of time, I've only included one map; however, if you'd like me to take any of my words and convert them into maps (for instance, climate maps), just email me or submit a follow-up question and I will be extremely happy to make them for you. I hope you enjoy this as much as I did!