Airport bus doesn't seem to start until after 06:30 (see timetable at http://www.empresafreire.com/html/ingles/seccion3b.htm) so I think your only option would be a taxi. One website I looked at quoted about 15 Euros from City Centre to the Airport but I don't know how recent that information was or whether there is an additional unsociable hours charge.
The most economical of course would be walking which would also save a night's accomodation
A taxi is not expensive, fifteen to twenty Euros, and the hospitalero usually will arrange to have one waiting for you at that hour, if you ask in advance. Nothing opens early at the airport, so arriving one hour before departure will be plenty. Even at that, you will be there before check-in opens or the cafe machines are warmed up. The seats in the terminal are quite uncomfortable.
It's a bit of a walk from city center. But there is a nice alburgue very close by, definitely within walking distance. Does anyone remember the name of it? It's right on the Camino before you get into Santiago, right across the way from the airport. Everyone passes it on their way into Santiago. There is a great bar/cafe there and many pilgrims stop for a rest before the last slog into Santiago. Maybe you could bus or walk there the day before and then you could just walk to the airport next morning.
Annie, are you thinking of the alberque in Arca? I stopped there to take a breather and make a restroom stop before the slog through the rains into Santiago. The Porta de Santiago in Arca (a little bit off the main route) has had good reports.
Well, I am sorry, I don't know the name. But we did not vary off the route into Santiago and we walked right past this place and stopped there for coffee and snacks. I could see the airport parking lot from there...through the trees.
The Bus is great, a short walk from tourist office to catch it 5-8mins, it's on the city map, but be careful cause Brierley has got it in wrong place!; if you caught 0610 bus you'll be there by 0640, peace of mind may mean you take a taxi.
The great thing about bus is you can relive (@ height) the last few hours of your Camino - only a few euros.
Thanks Annie and David, yes the bus is great. Took it from a dedicated airport stop on Rua Doutor Teixeiro a few blocks from Praza de Galicia, from where it stopped at the bus station en route to the airport. It's a wonderful feeling when you stand at a deserted early morning bus stop waiting and praying, to see that bus coming around the corner, on time!
I'm a big fan of the RENFE cercanias (commuter line) option. It leaves from Sol, you change in Nuevos Ministerios (an EASY change, unlike the metro line change at Nuevos Ministerios), and goes straight to the airport.