maybe you should actually,
you'll probably want to play it through twice anyway, because of the new game plus thing.http://uk.xbox360.ign.com/articles/119/1199706p2.html
there is the bit i mentioned about
Leading up to Batman: Arkham City, Rocksteady talked a lot about the Catwoman missions in the game -- that 10 percent of Arkham City was all about Selina Kyle. Well, it turns out that content isn't included on the disc. If you buy the game new, it comes with a one-time use code to download the Catwoman content. Used copies will most likely have the code snatched up and you'll need to buy the DLC off Xbox Live or the PlayStation Store.
If I can give you some advice, download and install the Catwoman missions before starting the game. Her four story missions, inserted throughout Batman: Arkham City, are a fun (albeit simple) change of pace, and literally change the opening of the game. I downloaded her stuff after beating the game the first time, and the integrated Catwoman stuff during my New Game Plus playthrough was way more enjoyable than just tackling her missions off the main menu. This is downloadable content that expands the mythos of the game as you're playing it. It explains events that are happening off camera.
Outside of the four missions, the Catwoman DLC gives you challenge maps for the feline and the ability to get the special Catwoman Riddler Trophies in Arkham City.