Basically, unless you are extremely brave, it is 'car essential' in that area.
Carrapateira has two buses a week, from Vila do Bispo not Sagres.
These run on Mondays and Fridays, leaving Vila do Bispo at 0805am on their way to Aljezur. You would have to get the 0740 bus from Sagres to Vila do Bispo to connect with it.
The return journey from Aljezur leaves also on Mondays and Fridays at 1330 if you want to to the long way way via Lagos.
Bus times Sagres - Lagos at http://www.algarvebus.info/012.htm
Bus times Lagos - Aljezur at http://www.algarvebus.info/036.htm
There is also a Rede-Expressos coach from Sagres to Lisbon via Carrapateira in summer but I believe it does not take local passengers.
That same coach runs every day from Carrapateira to Lisbon but where it stops there I don't know. It leaves at 1625 arriving in Lisbon at 2110. Rede Expressos has no sales office in Carrapateira so you would possibly have to book on line (http://www.rede-expressos.pt) in which case you need to have access to a printer. The nearest Rede Expressos ticket agent to Sagres is in Lagos.
I hope that helps. It is a very remote but very beautiful area.