If your flat earth has a somewhat normal gravity, whether artificial or by having something like a super dense ring at the edge that balances the normal center-of-mass direction, the atmosphere would just be present on both sides of the disk and around the edges.
There would likely be intense storms at the edge because of clashing jet streams, so to the inhabitants, the other side would be invisible. If the bottom side is also inhabited, an occasional unlucky explorer might mysteriously end up on the top side, thrown there by a storm.
From your edit, I assume that the "gravity" of the Universe overpowers that of the earth itself, causing anything that is not at the "bottom" of the universe to fall down. That of course leaves the question of how much air is reaching the bottom or is already at the bottom, and what else is there.
If the "bottom" is like an infinite flat plain that only receives some pockets of matter when a flat planet or gas cloud reaches the bottom, the air falling off the earth would spread out so thin it would not be noticeable, essentially lost forever.
If on the other hand the amount of air having reached the bottom of the Universe is big enough to create measurable pressure, the air could be reclaimed in some way at the bottom of the flat earth. That might be true if the bottom receives new air and water in balance with the natural dispersion on the bottom. Of course, that would also mean the earth gets new air and water falling on top of it.
A kind of stone with thin channels might absorb the water (and air dissolved in it) through capillary action, though I'm not sure it would work against this universal gravity.
Or there are world trees growing roots down to the bottom, where they absorb water and air through osmotic pressure, after which they evaporate/expel the water and air from their leaves as part of their metabolism.
Neither of them would seem to balance the loss of atmosphere from the sides if there's not some barrier there, though they would make up for some loss.
Since you have already handwaved away some major physical laws, you could perhaps declare that air pressure isn't a thing in that universe. So the air is not pushing itself to the edges, it only falls off when it some bit of it is pushed over the edge.
Finally, if the flat earth is artificial, it could have either some force field (solving all issues by handwaving again) or an air recycling system where the air falling off the sides is captured some way down along the the edge, filtered (to extract any intrepid explorers and their balloons) and transported to a mountain or otherwise in the middle of the disk's surface. This would result in a constant wind from the center of the disk to the edges, making any air travel one-way and very risky.
The problem with reclaiming air in any way is "what is out there past the edge of the planet?", because no regular pump or suction device is going to compete with a vacuum out there. The closest I can imagine is a combination of strong ionizing radiation and shaped magnetic fields to force the now charged particles into openings in the side of the disk.