I rent, but I don't see why this wouldn't be feasible.
Feasability is one issue, legalities and liabilities are another. You would have to get your landlord's permission to make any changes to the lock. You won't be able to install something like this without drilling holes into the door, which could open you up to the cost of replacing the door when you vacate.
And I would obviously always have the key with me at all times in case of failure.
The safety issue is about EXITING, not entering. Having a key in your hand will not help if the mechanism jams up, leaving you unable to quickly EXIT in the event of an emergency.
What kind of motors can you attach to an Arduino? Would a 9v battery suffice?
The Arduino cannot DIRECTLY drive ANY motor. You would need to either use a Motor Shield (made to piggyback on Arduino) or other Motor Controller to operate ANY motor via the Arduino. And no, a 9 volt battery would not work - it doesn't have the current capacity.
So at the least you would need the Arduino, the Motor Shield, a Wireless Shield, and XBee (or similar) wireless module, a hefty battery pack, and a gearmotor. Probably at least $200, and more later when you don't get your security deposit back from the landlord.