Hi!
Yes there is public transport between Faro Airport to Lagos or Alvor.
Getting to Lagos/Alvor from Faro Airport can be done by:
Bus - Faro Airport - Faro Bus Station
Bus - Faro Bus Station - Albufeira
Bus - Albufeira - Alvor/Lagos
Or
Bus - Faro Airport - Faro Bus Station
Train - Faro Train Station - Lagos
There is no train to Alvor. Descend at Portimão and take a taxi.
Again allow a transit time of up to 5 hours as these are local buses and train .
Note these services do not run hourly and forget buses/trains if you are arriving any time after 16:00 as public transport shuts done circa 19:00.
The cost will be in the order of Euros 15 per person I would think.
For general algarve public transport travel info see the following sites:
www.solpraiaferias.com/index.php?content=cheg_en
www.algarvebus.info
www.cp.pt
Taxis exist but can be pricey.
The 2007 Faro Airport Timetable gives a guide price of:
Faro Airport - Lagos (4 seater taxi) of
Euro 76.00 (Monday to Friday)
Euro 91.00 (All other times including week-ends)
There are several companies offering minibus shuttle services from faro Airport to Lagos and Alvor.
Have a look at these websites for starters:
www.low-cost-transfers.com
www.solpraiaferias.com/index.php?content=transfers&lang_opt=en
www.polotur.com