4 full days in Dublin

For a half day trip out of Dublin consider
Newgrange/Bru na Boinne

For a long day trip out of Dublin, or better an overnight, there's Belfast or Kilkenny.

If you've not been to Newgrange then that would be top of my list. Mary Gibbins tours are highly recommended .

The other place I'd recommend is the early Christian site of Glenddalough in the Wicklow Mountains to the south of Dublin.

There is a lot to see and do in Dublin if you haven’t explored the city before, but otherwise Newgrange would top my list also. I did a guided day trip that also included other sights such as Monasterboice and the Hill of Tara. You could also do a guided day trip to Glendalough. Rock of Cashel is worth seeing, but it’s fairly far south, and I’m not sure if there is a day trip that will go there from Dublin. With so few days, I don’t know that I would try to spend a night in another location as changing locations will eat into your sightseeing time, but both Belfast and Kilkenny would be easy as day trips as they are only about an hour by train. If you got an early enough start you would have enough time to see the sights.

How about this for a twist?

One of our traveling friends is dying to see the Cliffs of Moher. Our direct flight from Hartford gets in at about 5:30 in the morning. We could rent a car and drive across to the cliffs. Sightsee there and in Galway. Stay overnight in Galway and head back to Dublin the following day.
Alternatively, after seeing the cliffs, could we take a ferry to Aran Islands spend the day and the next there and take a ferry back and head to Dublin. Spend the last two days in Dublin.

Is that crazy?

You could take the bus directly from the Dublin airport to Galway, rather than renting a car. Much less stressful after no sleep the night before. (You’ll be landing just after midnight Hartford time.)
Then, take a guided tour of the cliffs from Galway. Spend the night, and then bus back to Dublin.

That’s what I would do, anyway.

Enjoy!

Re: your updated comment about Cliffs of Moher. I personally would not get off an overnight flight and get in a car that has driving controls on the opposite side of what you're used to. Then attempt to drive 3+ hours jetlagged. Not sure when you're going, but weather could be an issue with your updated proposal to see the cliffs then take a ferry to Aran Islands - depending on when you go, those ferries might not run (due to weather) and the cliff views could be obscured. I appreciate your traveling companion's desires, but I would focus on making the most of your very limited time on the east coast of Ireland. FWIW I regret not going to Newgrange, I'd definitely pick that for a day trip.

Good thoughts.
Mid April. Btw

Is that crazy?

In mid April - yes! Actually I've always found the Cliffs to be very over rated. Save them and the Aran Islands for another trip...

I agree about the cliffs and Aran, not a good idea with the limited time you have, and especially not a good idea to drive after an overnight flight, and especially on the left side of the road.

Dublin has plenty to occupy your time, see "Explore Europe" on this website if you want our host's advice. A day trip to Newgrange would be a good way to get into the country a little and see a unique world-class monument. I thought the Mary Gibbons tour, linked above, was very good and made the logistics easy. It does take pretty much the whole day, but includes the historic and scenic Hill of Tara as well.

Newgrange and Glendalough are two places that deserve serious consideration. Do some google reserch and see what you think. With regard to driving after an overnight flight, while probably jet lagged, on unfamilar ground, and doing it on the left side of the road, please don't. A lot of people (myself included) shouldn't be allowed near sharp instruments under those circumstances -- much less allowed to drive. Even a minor fender bender at the start of the trip will put a damper on the entire rest of the trip.

While driving after getting off a flight is not recommended, it’s not impossible. I did it on Wednesday this week. Got off a flight from Boston (before that two other flights) at 8:30am and drove even further-to Dingle. It was my first time driving on the left. I was alone and did fine. I then drove up to Cliffs of Moher and am now in Galway. Dublin next. It’s highway all the way to Galway for you, not a hard drive. I would go there first, find the place you are staying, take a rest or leave from there to drive down. Or you could get on a tour bus to go down. The roads are a little trickier heading directly to the Cliffs. They are only about an hour and a half from Galway. You might be able to go to the Aran Islands too but my understanding is weather could leave you stranded, be aware. Or make a day of it at the Cliffs. I followed Rick’s recommendation on hiking across the tops and loved it.

I'm glad Kathy had a good experience driving across Ireland after an overnight flight, but that remains riskier than doing it after a night's sleep. If you want to start your trip in Galway without driving there, take an Aircoach directly from DUB. You can get a car the next day for the cliffs, or take a tour to Aran. Then a train back to Dublin (city, not airport) for the rest of your time.

But, as I posted above, I suggest you focus your limited time on Dublin, take a day tour to Newgrange or Glendalough if you want, and skip the driving. Especially since you've been to the west coast already.

Appreciate the input. Agree- we'll avoid the long post-flight trip West... and focus on Dublin and surrounding sites.
With a direct flight from Hartford, it's easier to go to Dublin than to California!

I'd consider a day tour to Giant's Causeway and Belfast. Personally, Rock of Cashel wasn't a highlight for me.

Holy cow the Giant Causeway, and the Northern Ireland Coastline look amazing. As one of our travelers wanted to see the Cliffs of Moher, this area seems to be quite amazing as well.

too much to see !!

PS. is it easy to cross into Northern Ireland? with our USA passport?

Just as easy to cross into N. Ireland as is to cross a.State here. No passport control.