Hints for handling rust in shower drain
If you wish to actually de-rust the tub, you have a few options, most of which are slightly destructive, to remove the rust. 


Wire wheels or other powered grinding/sanding devices will be quickest, but remove the most material. 
Hand sanding/abrading will remove less material and may allow you to shape your result more easily.
Using an abrasive cleaning product will remove the least material of these manual processes, but may take considerable effort.
Electrolysis would remove only the rust, but likely involve some serious disassembly, not to mention voltages high enough to minimally cause pain (or worse). 


I recommend Bar Keepers Friend. Not affiliated in any way, we found it recently when trying to do some deep cleaning on a steel kitchen skillet, and have since tried it on more things than I care to remember. The latest is an electrical stove with three years of hard use and no interim cleanings. Within 5 minutes everything was off that should have been and the finish was unharmed.

If you were doing a more involved and personal restoration, I'd suggest electrolysis, but not for a rental unit. 

Lastly, you may want to consider requesting a fix from the landlord, as the problem may manifest again, given the installation of that recessed drain.