Hole Saw - This is a must if you want perfect holes. But if you are good at using a jigsaw, that can work too.
Ok. I dunno what tools you've got or how brave you are feeling.
I can see several approaches:
Kick in the door. Contrary to what some people have said, it's actually not that tough, though in some places people have pretty well fortified apartments. I dunno about your building. The basic trick is that you want to focus your body weight on your foot which should hit the door at the height of the lock mechanism, but a few inches towards the center. This will probably flex the door enough to pop it open, and/or break the doorjamb inside.
Cut a hole in the door big enough for the cat to run out. Maybe with a hole saw (you will need a drill) http://www.amazon.com/EazyPower-Hole-Saw-CornHole-Boards/dp/B000YC1XUK/ or a reciprocating saw http://www.amazon.com/DEWALT-DW304PK-Amp-Reciprocating-Saw/dp/B0002AJNQ0/
Locksmith. This is probably the easiest/cheapest/quietest, just pretend you are related to the dude and explain the approximate circumstances (e.g., "my uncle is in the hospital and we don't know where his keys are and we need to get the cat out").
Make a hole through the wall between your apartment and his. You could probably do this pretty quietly with a hand saw if you don't want anyone to know immediately, though ultimately the truth will be pretty obvious. You could probably do this pretty quickly with a reciprocating saw, but they're not quiet.
Attack the grate - I would think about the reciprocating saw again, or else drilling out the rivets that hold the grate together, they're probably weaker than the metal of the slats. You might also have some good luck using a jack (like for a car) or a pry bar to push the slats away from the wall where they're attached.