Is there a Batcave in Arkham City?
Solution 1:
No; Batman gets all his additional gear flown in.
Solution 2:
Actually there is a Batcave in Arkham City, although you can't access it through the story mode its just a challenge map in Riddler's Revenge which you you have to buy through DLC.
Solution 3:
No, Batman does not have a facility here.
Batman will either have additional equipment flown in or he will invent additional gear based on what he finds in various villain lairs.