The Oakwood Arms in Shannon are open for non residents breakfasts and not far from the Airport. Otherwise it depends on the route you plan for getting to Galway, straight up the M/N18 double back from the Gort (currently last) exit into town and O'Gradys or continue up take a left down to Coole park for a leg stretch wander then back up and mother Hubbards truck stop between there and Oranmore. If you take the coastal route then Randaddy's at Lahinch is a good place or a couple of spots in Ballyvaughan and Kinvara.
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Two rules for Ireland, Take your time & bring a sense of humour
Shannon airport has a restaurant called The Estuary Cafe above the check in area. It is open at 5 AM and serves a full Irish breakfast. The airport terminal is not very big and easy to get around.
In Adare there is the Dunraven Arms Hotel. There is a cafe in the visitor's center. The Golf Club at Adare Manor has nice food. I'm not sure if they are open for breakfast. A quick email should find out. There is a little cafe called The Good Room. I ate there twice and it was hit or miss.