Transportation from Rafik Hariri International Airport to Hamra, the neighborhood in which AUB is located, is relatively easy. Taxis are available at the main entrance of the Airport for $35.00 a ride. Also some of the suggested hotels may have pickup services from and to the airport for a fee.
For information about Lebanon and Beirut see http://www.lebanon-tourism.gov.lb/
If you wish to go around Beirut, there are taxis and shared taxis (service). Also there are many companies that can arrange for a taxi with a driver that can take you anywhere in Lebanon.