score:7
Cheapest is probably hitch hiking, but I am excluding that as an option. At least for the answer.
Train:
You can get detailed information in http://www.cp.pt/ for the train and in http://www.metro.pt/ for the subway company.
Bus:
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/
Conclusion:
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.
Additional information:
Additional resources:
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