From the bus station itself (Termibus) you can get to Bilbao/Bilbo Airport with Bizkaibus, at platform no. 12 just about 30 metres from where you get off the bus from Pamplona/Iruña, Vitoria/Gasteiz, Logroño, etc.
The journey from the bus station to the airport takes approximately 35 minutes and you pay for the ticket on the bus, Bizkaibus.
Consult times in bizkaia.eus