How to insulate an apartment you don't own?

Do you feel actual ‘drafts’, meaning air leakage through or around the windows and doors? That sounds like an issue that a landlord should be taking care of, though I’m not sure what sort of tenant rights you have where you live, or how amicable your relationship is with the landlord. May we ask , what is your climate zone? Your winter conditions may affect what we would recommend. For now your best starting place are these pages here - 

• Top Tips for Getting a House Ready for Winter

• Best Energy Saving Tips for Green Homes

For now, assuming a reasonably ‘cold’ climate and also running with the assumption of air leaks, you could do a lot to reduce that with some caulking or tape if you can identify clear locations where leaks exist. There is also the option of applying a plastic film with double-sided tape over the window, and maybe the door if it isn’t one you need to use for the winter.

I would check the weather stripping around the door, that may need to be replaced, which again, one would hope is something a landlord would take care of.

And if you really don’t care about the appearance, as a way to insulate it but not permanently and in a way that it could be removed, you could consider insulating it in a very temporary sort of manner by sticking some rigid foam boards (learn which are best here) to the wall with a few tiny nails, holes that small are easily repaired when you move out. That would go a long way to making it more comfortable. It means looking at a wall of your bedroom with sheet insulation against it, but maybe you can hang something over it to make it a bit more appealing.