Before you do anything else, go back and count (or re-count) your days and nights. Do you really have 7 FULL days in Ireland - not including your arrival day or your departure day? Remember, if your flight home departs from Dublin, you should spend your last night IN Dublin (rather than taking a chance you might miss your flight if you sleep the last night somewhere else and are counting on getting back to Dublin the day of your flight). Also, if you are arriving in Ireland from a long flight (from North America) you will probably arrive exhausted and sleep-deprived from the overnight flight and pre-travel business, so don't count your arrival day as really usable. Be honest with yourself about how much time you really have (if you are flying in from somewhere close, in western Europe, and your flight arrives in the morning, then your arrival day is more usable). But even 7 full days is a bit short for Ireland if you want to roam around to the west.
To your question: Yes, there's a bus from Dublin airport that goes straight to Galway. The bus is cheap, comfy (even has wifi), efficient, and is a great way to go. You can gain some efficiency if you simply get off your arrival flight, walk outside the terminal (short walk even with bags, easy to find). So yes, get on the bus to Galway at the airport, ride the bus, get off in Galway, check in to your accommodations, walk around for a bit to try and stay awake until dinner, then sleep deeply - your Ireland trip actually begins the next morning. You need to be in Dublin anyway for your flight home, so you'll just skip Dublin initially and you'll be there after your quick trip out to the west. This is an efficient way to begin your trip.
Ireland is great. Have fun.