Getting to Bilbao and San Sebastian
The closest airport to Azurmendi is Bilbao and it's also the nearest Spanish airport to San Sebastian.
We flew over with British Airways and personally felt that their flight times worked the best. The 8.30am flight from Heathrow goes most days - easily getting you to Bilbao in time for an early lunch sitting at Azurmendi and if you can't take much time off work, an evening flight at 5.15pm or 8pm would get you back the same day if you wanted. The restaurant itself is only about 15-20 minutes drive from the airport, so it's very easy to get a taxi there and back.
Getting to San Sebastian from Bilbao airport is easy - there are regular coach services going direct from the airport to the city and it takes about an hour and a quarter. If, however you'd like to cross something else off the list head into Bilbao bus station and get the bus from there, as the coach to San Sebastian goes up past the Guggenheim.
You can, also get to Bilbao by flying into Biarritz which is a slightly shorter journey time into Bilbao although the flight times aren't as good, particularly if you're just coming here for a weekend.