Bus Info: See Cubsur's: http://www.algarvebus.info
Fado music in the Algarve off-season might be a bit difficult. Its more a Lisbon and Coimbra Thing.
The "O Papagaio" restaurant in Espiche outside Lagos, in the Western Algarve, does offer Fado sometimes but I have not been out there this year.
See: http://bit.ly/9AH3lD
Although its a bit of a way from Faro, I would suggest you have a look at Lagos, Burgau and Salema in the Western Algarve. Good beaches and good public transport links.