Bathtub drain & overflow
As a total novice DIYer I would say that this job is beyond what you are prepared to do. The house is an old rental so the previous work is old and most likely shoddy. Even if you were able to change the components successfully, you may cause more trouble somewhere else from wiggling the pipes around.

But if you must:
The plywood you removed can cover the hole you make in the drywall so cut with a drywall saw, but don't go deep. Just one inch max. And if you feel something that doesn't feel like drywall stop cutting and make a small hole to investigate. There are pipes in there and though there shouldn't be wires, there may be. Once you have a hole large enough to take a picture, snap a few and post it here. If we can help, great, if not replace the plywood and call your landlord because the overflow "fell off" and your worried about a leak. Oh also, is this a basement washroom?