Lighting – How to add a ceiling light fixture to the apartment

If you're looking to add a light to an existing ceiling fan, making it a combination fan/light, this will only work if (1) the fan is designed to accept an optional light add-on and (2) you use the light add-on appropriate to your fan. If you don't know what type of fan you have, this will be difficult to find, but you could start by turning off the power and seeing if you can detach a part on the bottom of the fan. If this reveals some wires or wiring terminals you may be in luck; measure any attachment points and look for an add-on from the same manufacturer that would fit.

If you're talking about just swapping out the whole fixture, you could relatively easily replace the existing ceiling fans or light fixtures with new fans or fixtures. Most fixtures are designed to attach to standard ceiling boxes. These sorts of changes can also be easily undone (remove your new fixture and reattach the old one) if needed, which may be useful in your situation. Note that if you're attaching a ceiling fan, you need to be sure the box you're attaching to is rated for fan use (i.e. has a stronger attachment to framing to support the weight/motion of the fan).

Do you have a picture of the cover? It seems very likely that all your problem simply relate to an overheating. Is there a chance for a better air flow (thermal ventilation)?