If you do depend on public transport to get to Alcoutim, you can do this:
From VILA REAL DE SANTO ANTONIO take the bus that leaves at 5:10 PM from the main bus station (€3,10 one way).
There is a bus going back from Alcoutim at 7:45 AM.
From LISBON take the Expressos bus to BALURCOS (CRUZAMENTO). From there on, however, you have to organize your own transport, as it is still about 7 km to the village. Not really recommended
There is a bus from MERTOLA going to Alcoutim as well.
Car or boat
The best way to get to Alcoutim is by car or by boat. Public transportation is not that well-developed.
A tip: if you want to take the most scenic route to Alcoutim, take the motorway to BEJA and MERTOLA and take the exit FOZ DE ODELEITE rather than Alcoutim. Turn right at the end of the exit and then left. Drive to Foz and just follow the Guadiana upstream...