The nearest reasonable hotel to Faro Airport is the Faro Ibis. It's about 3kms from the airport.
See: http://goo.gl/78Bd0
You will still need a taxi to get there and back. Maybe the car hire company can deliver the car to you at the IBIS the next morning?
It does have a bar.
Have stayed there and it is a typical IBIS. Reasonably good quality for a reasonable price as one would expect from an Accor Group Hotel.
Otherwise you'll have to head into town to the more costly Hotel Eva or the Hotel Faro. There are Pensões but none as far as I can remember with bars.
