Cheapest is probably hitch hiking, but I am excluding that as an option. At least for the answer.
- The fastest is probably the train. The fastest connection takes 3h40 min. It goes from Mexilhoeira Grande to Tunes. There you change and it goes directly to Lisbon, to Oriente. Oriente station is the closest train station to the airport. From there you get to the airport by subway (3 stations away). This should cost you around 27euro (all transport tickets including the subway). I don't know how far you are from the starting station. Mexilhoeira Grande is probably the closest train station to you but you'll have to check. This train is at 6h24min in the morning. All the other trains take around 4h.
You can get detailed information in http://www.cp.pt/ for the train and in http://www.metro.pt/ for the subway company.
- You can get the bus directly in Alvor. It also goes to Oriente in Lisbon. From there you get to the airport by subway (3 stations away). The total cost should be around 23euro (all tickets including the subway). The total time on the bus is 4h15min
You can get the bus schedule here http://www.renex.pt/Files/Horarios_SUL.pdf The bus website is http://www.renex.pt/
You can get additional information about the subway here http://www.metro.pt/
Unless you have a good reason to go that early and spend less time travelling, I would probably go by bus. It's more convenient (no changes) and it doesn't take much more time.
- When arriving to Oriente in alternative you can get a taxi. Depending on the number of persons traveling it can easily get cheaper and a bit faster than subway to arrive to the airport.
- Lisbon airport has 1 terminal for arrivals (Terminal 1) and 2 terminals for departures (Terminal 1 and Terminal 2). The subway station in the airport is at Terminal 1. There is a shuttle bus every 15 min to terminal 2. The trip takes around 5 min. As a thumb rule low cost company's depart from terminal 2 (I would still advise you to check just in case).
When you arrive to Oriente the station is easily recognizable for its architecture:
Map of Lisbon subway system and a portion of the train system.You can clearly find Oriente metro/train station and easily see the distance to the airport station (aeroporto is the name of the station).
You can easily see the distance from Oriente to the airport by car in Google Maps