The best bet is the Jumbo hypermarket in Faro.
See this link:
http://www.jumbo.pt/Frontoffice/ContentPages/Store.aspx?StoreId=033
They are open till 24:00.
To get there from the airport do the following:
Get on the main road out of the airport and follow the signs for Faro.
After about 3kms you will see an off-ramp to your right for Faro.
Take it!
Watch for the traffic merging from the left as you enter the N125 dual carriageway.
Continue about 1,5km and you will see on your left the shopping center "Forum Algarve".
You will then immediately come to a roundabout.
Go 270º around the roundabout and take the entrance to the shopping centre.
Park above or underground.
You will find Jumbo there.
On leaving the shopping centre follow the signs for Lisbon which will eventually bring you to the Via de Infante Motorway and follow the signs for Albufeira, Portimão, Lagos.
