Yes, it seems that Arkham City assumes that you have played Arkham Asylum as it doesn't take the time to explain combat or gadgets. AA does a great job of slowly getting you aquainted with the various gadgets and how to use them.
I just completed Batman AA GOTY recently, and will be recieving AC in the mail soon. I would highly suggest playing AA first, it is an amazing game even for being 2 years old now. But I really think you could get away with playing AC first if you wanted to.