The fastest and most direct way from Coimbra to Lisbon Airport is by train. There are no direct buses from Coimbra to Lisbon Airport.
By Train:
1) Take a train from Coimbra-B to Lisboa Gare Oriente station.
2) Descend to the ground floor and head towards the bus station.
3) Find the bus stop for the Aeroshuttle (Line 96). (See: http://www.carris.pt/pt/autocarro/96/descendente)
(It was by the taxi rank last time I was there.)
4) This will get you direct to the Airport.
If you prefer a bus you would then have to get a Rede Expressos bus (http://www.rede-expressos.pt/) from Coimbra to Lisbon Sete Rios Bus Station and then the Aeroshuttle in the other direction back to the airport.
