A couple of things to bear in mind.
1) When leaving the airport head north in the direction of the A1 motorway to Porto, then follow the signs for the Ponte Vasco da Gama, A2, Algarve. There is usually much less traffic on this crossing over the Tejo and it avoids getting tangled up on the various ring routes around Lisbonand over the 25 de Abril bridge.
2) There are normal pay tolls on the A2 South and A22 until you get to the Albufeira off-ramp. Thereafter the A22 turns into electronic toll land, with all that entails.
I would probably personally come off the A2 at Messines and follow the N124 to Silves, then the N124-1 to Lagoa and thence onto Carvoeiro. A lot less traffic on the route, albeit single lane each way. However, it avoids the tolls. Might take 15 minutes longer.
