Audio problem: When Batman types on the computer in Dr. Young's office, the clacking sounds are woefully out of synch with his hand movements, not to mention his fingers never actually touch the keyboard.
More trivia for Batman: Arkham Asylum
Question: In the comic books, animated series, etc, was there a passage connecting the facility to the bat cave like there was in the game? I thought that the bat cave was further away, like under Wayne Manor.
Answer: It's not the actual Batcave, it's a secondary location Batman set up in case of emergency. Oracle actually makes a comment along those lines the first time you access it.
Answer: To add to the answer, Batman tells Oracle that he found the cave after saving an inmate from committing suicide and had secretly been moving supplies to the cave after discovering its location.