Realistically, as mentioned by others, Faro and Lisbon are the only feasible airports for the Algarve.
From Ireland I think you are restricted to:
Ryanair
Dublin to Faro
Shannon to Faro
Aer Lingus
Belfast to Faro
Cork to Faro
Dublin to Faro
Dublin to Lisbon
www.aerarann.com
Waterford to Faro
Galway to Faro
easyjet.com
Belfast to Faro
That's all the current scheduled services I can find announced so far for this summer. There are always the possibility of charters etc
