By far your best bet for actually birding the premises is to check the Family Viewing Area at the end of old Golf Course Road, as well as any open areas en route with binoculars. 

Please note this area is only accessible by car and is basically impossible to access by foot or public transit.  Be careful not to drive or walk off the road as this will draw the attention of airport security.   

Loggerhead Shrike, Crested Caracara, Northern Bobwhite

Not just a place to be stuck in transit or on layovers, Austin-Bergstrom International Airport can yield good open-country species during the colder months and is a good open space for spotting migrating flocks during spring and fall. The caveat is that you have to get away from the terminal and have your own ground transportation, and this is so close to Hornsby Bend that serious birders with more time on their hands should probably go there instead.

Primarily of interest to patch birders looking for open-country birds and birders who double as planespotters.