That's what was meant by the Northern route.
Westport is served by train (indirectly from Galway), and of course Derry to Portrush is, and there is an hourly bus service from Drogheda to Trim and 2 hourly from Trim to Dublin.
From Galway to Westport anyone who wanted to do the route by public transport would take the bus, from Westport to Derry they would drop back from Westport to Claremorris on the train, then the bus to Derry.
So it is possible by public transport with relative ease. But that wasn't your question, so it is a red herring other than placing a point of reference for future readers.
Enterprise and Europcar both have offices in Galway. Both would need to know that you were routeing via the Province, but either should be able to provide a suitable vehicle.