How can I fix the Batsuit?

Short of switching into another suit (gotten through a preorder bonus, or through DLC), there is no way to repair the suit.

Beating the game unlocks the option to change into other suits, however, you can only change costumes before you start playing. When you choose either "Continue Story" or "Continue New Game +" you get the option to change your suit. You will also get the option to change suits in the challenge modes.