You probably don't need anything as fancy as tsl's setup either. I assume you have a wall in the apartment where you can lean your bike. Just make sure you have something under the bike. This could even be as cheap as some 6 mil plastic.
Before you bring the bike upstairs, bounce the tires a half dozen times to shake off snow. Then wipe down the frame and drive train with an old rag (you might have to carry this with you or stash it somewhere so you don't have to make a trip upstairs.) You might have torun the rag under the fenders too.
That should take care of the majority of the grime and the rest can drip off unto the plastic.