Dublin Airport - Rent a Car?

If upon arrival you are going into Dublin, then there’s no reason to get the car. Take other transportation into town and pick up the car as you’re leaving. You shouldn’t need a car in the city. Since you have an early departure, maybe spend the last night closer to the airport. Check the hours the rental car office is open as it may not be open so late at night or early in the morning. I assume if it’s closed, you can drop the car off and deposit keys someplace.

You really don't want (nor do you need) a car in Dublin.
If you're heading to Galway anyway why not make this a little easier on yourself and take the train, then pick up your car in Galway. It's a lot easier getting used to driving on the left on country roads (the N roads) than it is in the city. You could reverse the process for your return to Dublin for your flight home and thus save yourself the aggravation of ever coming close to Dublin in a car. Plus, you won't want the hassle of an early morning car return before your flight.
Specify a car with automatic transmission - the default rental in Ireland is a manual (ie left hand shifter) which is an extra fun factor that you really don't need.
Use the street view feature on google maps to preview your routes - really helps me to have seen the sight picture on the roads before I have to tackle them at speed - might help you too.

We picked up our car at the airport on our way out of Dublin. As others have said you do not need a car in Dublin. You can take the airlink bus into the city when you get there and then back out to the airport to pick up the car. I would also drop off the car at the airport the night before your 5 am flight. There are major highways around and getting ur of the airport which makes it easier to practice driving on the left before you hit some the country roads.

Since you aren't used to driving on the left I would strongly advise against picking up your car in the city. On our first trip to Ireland we chose to pickup the car downtown after our stay in Dublin and my husband, who is an excellent driver, ended up clipping the passenger-side mirror within the first ten minutes. You would be much better off taking a bus or cab back to the airport and continuing your travels from there, or, as a previous poster suggested, taking a train to Galway and renting a car there. At the end of that trip we chose to stay at the Holiday Inn Express near the Dublin airport so we could drop off the car the night before our flight home. HIE had a free shuttle to and from the airport so it was easy and stress free.

Save yourself a ton of stress and just say no. Seriously, the worst thing we ever did in Europe was rent a car in Ireland. It was unbelievably stressful, difficult, etc. if you must however it is a tiny bit easier driving in the countryside as opposed to driving in Dublin.

I would recommend taking a taxi to the airport after your couple of days in Dublin and leaving from the airport. Automobiles are generally liabilities, not assets, in cities and especially in old European cities.

While driving on the wrong side of the road takes some getting used to, as long as you get a small enough car and are careful you should be fine. It helps to have a passenger on the left telling you to get the heck over, because for the first day or so it can be difficult to gauge where to place the car on the road. It helps to be patient and be aware of if you should pull off the road to let oncoming traffic pass on narrow roads. It depends on the dynamics and availability of space to pull over -- sometimes it's better for an oncoming car to pull over, sometimes it's better for you to pull over for them.

Thank you everyone for the helpful feedback. I will not be renting from Dublin :)