Dublin- renting a car a few days after arrival

I agree that you won't need a car in Dublin. The city's famous for streets changing names every few blocks, and driving there is a little confusing.
It's not terribly far from the airport into the city center, so you could catch a cab. Bus service is also good on main thoroughfares. We stayed at the Hilton out by the airport, and they didn't even have a shuttle.
Good luck driving on the "other" side of the road. I've driven right hand cars many times, so it wasn't a problem for me. But the backroads are often quite rough and they can have ditches, walls and bushes right up against the driving lanes causing difficult driving conditions. That's the truth.
If you're coming into Dublin and flying out, consider staying in an airport located hotel.
And don't croak when you are required to spring for CDW on the rental car as it's mandatory in Ireland. Make sure it's priced reasonably when you make your rental reservation.

Nothing beats the door to door service of a taxi, especially if you're horsing around luggage, and for two of you (or more) the cost isn't that terribly much greater than taking the Airlink shuttle.
The default rental is a manual so you'll need to specify an automatic at booking. The airport concessions have a greater selection of vehicles so getting the automatic you've booked is usually not an issue. Can sometimes be hit or miss from the pickup points in the city.
The airport lot is located about a mile from the airport itself, so you'll save yourself a bit of time and a lot of aggravation if you take a taxi directly to it on the morning of your departure.
Finally, the airport concessions open around 5 AM if getting an early start is a consideration.

For us it worked great to get the car at the airport and then after our two weeks in Ireland return the car to the airport. We paid extra for an automatic and did not regret the extra cash spent. We took the bus into Dublin and then took a taxi on our way back to the airport. It was around $30 for two of us from the Temple Bar area. The taxi picked us up at 5 AM because we had an early flight and our B&B contacted the taxi the night before we left and scheduled the pickup and the guy was 10 minutes early, perfect! The bus to Temple Bar from the airport was cheap but after going that route once I would now spend the extra money for a taxi both ways. The bus was extremely crowded and stopped at numerous stops. I wanted to hit the B&B,drop our stuff, and hit the town and the delays were a pain. We were only about a quarter of a mile from the bus stop but it was a stressful to find our to trek with our suitcases to the B&B, I would spring for the taxi and get dropped off at the front door of the B&B. Your expense will be a bit more as you are going into the city center twice but I saw the traffic in Dublin city center and it would not have been easy for me. I guess on your return to Dublin you could stay out near the airport and many folks do that, we didn't go that route so you will need others to give advice on that. We loved the south and west Ireland!!!

We flew into Dublin, and despite our plan to bus to our B&B, we were just too jet-lagged to deal with it so we took a taxi--a good decision. After our three night stay we picked up our rental car downtown--a bad decision! Never having driven on the left before, navigating Dublin traffic was a nightmare! I heartly recommend taking a cab back to the airport to pick up your car. For the trip home we chose to turn in the car and spend the night at a hotel close to the airport; and yes, they did have a shuttle service to the airport.

This is a good time to pay for a taxi, imo. It's not a crazy amount of money, and after a flight from the US your coping skills may be diminished ;-).....while you're in the taxi you can chat up the driver for some hot tips on left side driving.

For us it worked out to fly open jaw, coming into Dublin, picking up a car in town a few days later on the way out of Dublin, and then turning the car in at Cork Airport the day we left-super easy. This may also work for your itinerary into/out of Shannon.

We rented through the Budget location in Drumcondra, just a mile or two N of city center-via Auto Europe IIRC. Our B&B host gave us great directions the day we left to easily find the highway West. (The fact that I took a wrong turn that cost us forty five minutes at rush hour is a story for another day ;-) )

Have a great trip.