The only option for getting by public transport from Tarifa to Lagos is by Bus via Seville. Note that between October and April, the direct buses from Seville to Lagos only operate on wekk-days. (Not week-ends nor public holidays.)
Timetable for coaches between Tarifa and Seville can be found here:
http://www.tgcomes.es/horarios_33_english.htm
Timetables for the DAMAS/EVA service can be found here
at Cubsur's excellent site:
http://thomasallen.home.sapo.pt/travel/001.htm
