In my rented apt., the bathroom sink drain looks old and wobbly.

sinks are built separate to the drain which means YES the drain can be replaced---some are easy everyday parts----some are special and require the ones made for just that sink!!! The special ones are more costly which I would think rental properties would avoid due to added expense when repair is needed!!! Take a picture (several best) with your phone and head to your local home center!!! Ace / Lowes/ Home Depot/ ETC.

Contact the owner or management company. They should be responsible for that repair. If they refuse, ask if you can do it yourself. Go to a home improvement store like Lowes, Menards or Home Depot and they can help you with what you need and how to do it.

Call the landlord.

Yes, the sink drain can be replaced with a new one. Use a pipe wrench to loosen the drain pipe and release the ring that holds it tight to the sink. Check the condition of the rest of the drain since the P trap often corrodes as well. Clean the sink drain hole where the flange sat. Insert new drain . You'll need plumbers putty to seal the joint between sink and the drain. Compression fittings work on the drain to P trap connection.

Don't forget the drain plug assembly.

Not sure what you mean by "Wobby" but you can go to any of the big box stores and get a new sink drain and its easy to install. However, I would talk to your manager before you make any changes to your sink. He/She may just take care of it and you will not be responsible. Most apt. use all metal drains that last longer than PVC types.

If it is wobbly, ask the manager to repair it. You do not have to do this yourself.