Trip to San Sebastián from the Bilbao airport:
The company Pesa has direct buses from the Bilbao airport to the bus station in San Sebastián (1 hr. 20 min., 16.85€). You buy the ticket on the bus. It leaves every hour from 07.45 until 23.45. The stop is just outside the airport, down the stairs on the right (follow the signs for "Bus").
A cheaper alternative, though more complicated (and for this reason we don't recommend it), is to take the Bizkaibus bus to the Bilbao bus station (35 minutes, 1.40€, you buy the ticket on the bus, last stop) and from here take the Alsa bus to San Sebastián (1hr., 6.47€, buy the ticket in ticket office).
It's a 15-minute walk from the San Sebastián bus station to the city centre and 20-25 minutes to the Old City. From Avenida de Sancho el Sabio, you can take bus 28 or 26, which goes towards the Boulevard making several stops in the centre (15 minute 1.65€). If you prefer or are travelling as a group, you can take a taxi to your hotel for around 5-6€.