It is not necessary to check with the airline on the visa. The UAE is very clear on their list of countries that do not require a visa (note - they call it a "visa on arrival" but it is free and really just a passport stamp.
Assuming you do not need a visa, Abu Dhabi immigration and customs usually clear quickly. Now, even by taxi they are a solid 45 minutes from the city so with buses be very careful of your times. I know it would be more expensive, but perhaps you would want to negotiate with a taxi driver to stay with you and wait for you to make this worthwhile.
I assume that if he has a stop over in Abu Dhabi it is Etihad? I don't believe Etihad does tours of Abu Dhabi and if they do - I very much doubt they do them at night.
I would probably go to the corniche. There isn't going to be a lot to do; even in the day - but the corniche will be full of people and is always fun to just walk. I think with this short time you will be able to view a bit of the city and just enjoy the ambiance.