Just returned from Santiago. The bus at the main station has daily Santiago to Finisterre runs. Bus to airport leaves from same terminal. I very much doubt there is a direct run. Timing is the issue since the Santiago-Finsterre run is about 2.5 hours. Depends on when you need to be at the airport
I don´t believe there is a direct route but it couldn´t be easier. Take Fisterra to A Coruña bus (Vasquez), change in Baio for the Aucasa bus (check for links "enlaces" as some have been discontinued with the fall schedules). At the bus station there are airport buses every 30 minutes: three euros.
Do note: the Airport bus runs from a stop across the dual carriageway from the bus station main entrance,there is a footbridge, and not from the bus station. Also the service is hourly on Saturdays, Sundays and Festivos.