Unless you're very lucky, it's unlikely that you'll find anywhere posting on this site who has actually paid by cash for short-term car hire at Faro Airport. Your best way of finding out the requirements of each of the car hire companies is probably to phone them up. (International calls aren't particularly expensive these days, and car rental staff at airports usually speak English).
Here's a list of the relevant phone numbers. (Put '00' in front of them to call from the UK):
http://www.ana.pt/por...nu=184927&cboui=95632
Chris