Bus No 3247 leaves from the airport to central Bilbao every 15 minutes (from 6:15 to 00:00 every 15 minutes). On the way to the airpot, the same bus leaves from Termibus (San Mamés) every 15 minutes (from 5:15 to 22:00 every 15 minutes.). Please check with the Bizkaibus web site for possible changes and updates.
A taxi from the airport to the center of Bilbao costs approximately 30 euro. There are several radio taxi companies (for example, taxibilbao.org/ or teletaxibilbao.com/en/.)