Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.
I know some buses that goes in that direction start at the bus station in Santiago, and make their first stop at the airport. Then there are buses that does not. Check the itinerary on the ALSA website (when you reserve there is a way to see all the stops this bus makes, with times).
If you need to travel into town, you need to take the bus (taxi if you are in a hurry) to the bus station in town. It is about a 15 minute taxi ride. I am not sure if the buses from the airport go to the bus station or if they drop you off at Plaza Galicia (more down-town). If it drops you off down town, you have a 15 minute walk back up to take your ALSA bus.
The taxi rate is a fixed price from the airport, about 15 euros (don't remember exactly).