There might be no problems at all.
Remember that people all over this world once thought the world was flat. Living on Earth, it's hard to tell the world isn't flat in everyday life.
Let's say the world is a very large flat plane -- a disk or something like that. Gravity pulls everything down in the same direction. The plane itself is held up by something reaching infinitely far down. It could be dirt and rock all the way down, or an endless stack of turtles, or infinite stilts -- anything will do. The sun, moon, and stars are all weightless and circle around the world, very far from the plane. Air fills the universe up to the level of the plain and just a little bit higher (to be the sky).
You can't see an infinite distance. Air itself isn't completely transparent, which is why distance mountains look bluish. At a great enough distance, things will just look sky blue. Mountains and other terrain features will also get in the way.
The sun isn't any closer to different parts of the world. Rather, it is closer, but so imperceptibly that it makes no difference. Remember, the distance from the earth to the sun is much bigger than the width of the earth. Assume the same for your flat world.
Seasons could still happen. The sun's path around the world could tilt north/south over the course of the year, just like the sun appears to do in our world. If the center of the sun's orbit is actually below sea level, and its orbit tilts far enough, you could have long periods of dark like at the poles on earth.
The sea of air could have swirls and eddies in it. As the sun heats the great sea of air around the world, it could have pressure differences just like on earth, leading to winds. Cooling happens as the air is in the shade of the world.
Magnetism could still direct compasses. If you're already assuming gravity points down the same way for everyone, it's not too hard to assume that magnets always have a force making them point north.
At the edge of the world, as long as you have mountains keeping the sea in, you won't have any problem with losing the world's water. Then again, it might be that the regions near the edge are completely dry, because their water has drained away. And it's only a matter of centuries till the whole world has gone dry.