The only daily connecting bus service is one that leaves Lisbon Gare Oriente Bus Station at 09:15 arriving Vigo 17:30.
However, it is possible to do the trip by train:
There is an Alpha-Pendular Express train that leaves Lisbon Gare Oriente at 14:09 arriving at Porto Campanha Station at 16:46. There is a train leaving Porto Campanha Station at 17:55 arriving Vigo at 22:12 Spanish Time.
You can get a the Carris Aeroshuttle bus to Gare Oriente just outside the airport arrivals level. This leaves every 30 minutes between 07:00 and 23:00 and goes directly to the airport for a cost of +- € 3.00
Alternatively, take a taxi to Gare Oriente. Its only 10 minutes away.
You should try to get to the station as earl as possible to get a ticket.
