Dublin, Belfast, Galway, and Cork - doable in 13 days/12 nights?

How many nights will you be in Ireland? Are you flying out of Dublin? If so, sleep in Dublin at the end of your trip.

At the Dublin airport take the Aircoach bus to Galway (3h 15m) or Cork (3h 30m): https://www.aircoach.ie/. You will need to rent a car in one of these cities when you’re ready to move on. Otherwise, you’ll spend a lot of time on buses that involves transfers.
Between Dublin and Belfast take a direct train (2h 15m): https://www.irishrail.ie/en-ie/.

Is it 14 days or 14 nights? Public transport can be slow in Eire.

Thanks for the clarifying question: it is indeed 12 nights that we'll be staying (and on further looking I updated that there will be 13 days; one of the days on the ticket schedule will be for traveling - I'm definitely out of the loop of travel lately!)

Apart from Aircoach there are two other operators to Galway from Dublin Airport, similarly to Cork. Aircoach must do a good promotion job in the US the way they are the only operator most people know of.

It is easy to get between the cities by train or bus.

Contrary to popular misconceptions you don't always have to go back to Dublin to go from Galway to Cork. You can switch trains at Portarlington and Portlaoise where the two lines have a shared route, or even route down the west coast to Limerick ( maybe overnighting there) then Cork.
It is about 4 hours from Galway to Cork by train on the better connections. Cobh is an easy day or half day side trip from Cork with frequent commuter rail service.
If you had to cut one out then cut Galway (if you are not doing the Aran Islands on this trip)- saving the west coast for a different dedicated trip.

My calculations always show it is cheaper (and faster) to rent a car when leaving Dublin. Taking a bus or train to somewhere and renting a car a few days later saves you a few days rent but adds a one-way fee, which is a similar amount. It makes some sense if you're heading somewhere on the day of your arrival because of your jetlag (and driving don't mix well) or if you're happy visiting only your target cities (and things like the Aran Islands, Cobh) then don't get a car. A car is faster than the fastest train and allows you to make interim stops to see things that aren't in the cities.

Trying to visit your entire list is probably too much for 12 nights. Without knowing what you like to do and see, recommendations are worthless. Spending limited time in the four largest cities isn't something that we'd enjoy, although they are are each worth a visit. We've spent nine weeks in Ireland and haven't ever been to Cork, although our next three week trip includes Cork. And we've never stayed more than two nights anywhere besides Dublin. My point it that there is LOTS to see and do in Ireland, and most of it is difficult without a car.

If you only have 11 days on the ground then I think you should remove the furthest away location, Cork, and stick with splitting your time between Dublin, Galway and Belfast. There are many guided day trips you can take from each of these three cities that will get you to many interesting places. You won’t see everything, but you only have 11 days so that wasn’t going to happen anyway. Also, you can fly into Dublin, and out of Belfast which will save you some time. The less you change locations, the more you will see as checking out of one hotel, taking a train or bus, finding the next hotel and checking in can end up using up half of a day. So limit the number of places you stay. Take a look at the various day trip options available in each city, if that is what you want to do, and that will give you an idea of how much time you need in each place.