Well, not JUST for this game...
I'm looking for a video card.  Right now, I only have the Radeon 4200 on the motherboard.  I can somewhat play this game, but it's slow.  GTA IV is worse.  Batman, I can at least get looking halfway decent or running somewhat smooth.  GTA is still ugly and still slow.  So, I wanna go ahead and get a proper video card.  Still having trouble deciding on one, though.  In my inexperience, I was wanting to go with ATI, but I know Arkham Asylum has PhyxX effects that pretty much only work on NVIDIA cards.  Would it be worth getting a card that has PhysX?  How many other games have PhysX?
By the way, right now, I only have an old 17" CRT, and I probably won't get a monitor bigger than 22" or so.

Not a lot of games have PhysX, and personally I don't really think it adds anything to the game.  
 
But any way, I recommend the ATI HD4850. It'll play any modern game with no problems and it's the best value-for-money graphics card out there as far as I know. I was gonna get it myself, but it's out of stock everywhere here, so I added a little more and got a HD5750, since it's just a little better and has DX11 too.
 
Just make sure you have a decent power supply.

" Not a lot of games have PhysX, and personally I don't really think it adds anything to the game.   But any way, I recommend the ATI HD4850. It'll play any modern game with no problems and it's the best value-for-money graphics card out there as far as I know. I was gonna get it myself, but it's out of stock everywhere here, so I added a little more and got a HD5750, since it's just a little better and has DX11 too. Just make sure you have a decent power supply. "

It does use physics, like walls and debri flying around, papers flying around, bugs on walls and a few neat effects, however you can live without it. Your card is too slow anyways and when there is physics, it really slows down the game with nvidia cards. Your best bet instead of investing in a new phsyics card is to get a new card, maybe an nvidia card instead.

Unless you find it for cheap, a PhysX card is useless. If your mobo supports SLI, an 9800GT costs $100 and does a better job.

In the case of Batman:AA, it depends on whether you like interactive smoke/fog (though UT2003 had that like eight years ago) and paper flying around somewhat realistically. The bugs on the wall are there without PhysX, too - they are not done with PhysX. The missing fog is probably the biggest difference. I played twice through the game with PhysX off (too much drain on performance) and I didn't really miss it. 
 
So, no. Don't buy another card for PhysX allone.