Santiago-Sarria direct

Yes, taxi. The bus connects through Lugo from the airport.

Empresa Freire provides bus service between Santiago and Lugo. It stops at the airport enroute to Lugo bus station.

Monbus provides bus service from Lugo bus station to Sarria.

The other option is trains which require getting from the airport to the train station. The trains require a transfer at A Coruna, Ourense, or Monforte de Lemos.

Of course a taxi is possible but its 1oo km from the airport to Sarria ... I'd call that very expensive.

Airport Santiago
http://www.aena-aeropuertos.es/csee/Satellite/Aeropuerto-Santiago/en/

Empresa Freire
http://www.empresafreire.com/html/castelan/seccion0.php

Monbus
http://www.monbus.es/index2.htm

Renfe
http://www.renfe.com/

Sarria Concello Transport page
http://www.sarria.es/index.php?option=com_content&view=article&id=109:transportes-y-horarios&catid=60:localizacion&Itemid=132&lang=gl

If you decide to do the "taxi" thing contact José Luis Vizcaíno <xacotrans@gmail.com>. My wife took a taxi direct from the airport last fall. She said it was worth it after the three flights. If you can find someone to share the total is a little over 100 euros.