Round trip: Dún Laoghaire to Dublin

For public transport, it looks like there is both a train and bus. The train is faster.

I am not totally certain as to where your cruise ship docks in Dun Laoghaire, but it’s a relatively small town. The train referenced by the previous responder would be DART, the local light rail. A commuter line. It’s like a bus, just show up and board. The train runs every 10 minutes or so, and will take you right into centre Dublin in 15-20 minutes. Several centre city stops, Tara, Pearce, Connolly are all central. Very convenient and cheap. Round trip from Dun Laoghaire is about €5. Plenty of buses also, to various parts of Dublin.

Essentially, Dun Laoghaire is a Dublin suburb. Lovely town.

The ship doesn't dock at Dun Laoghaire. It anchors off with tenders ashore.

That means you have no idea what time you will get ashore-- which makes pre booking anything impossible.
But as others have said there is very frequent bus and train service.

Also check what time you have to be back for last tenders (well before departure time), and check you have phone service in case the weather worsens, and you have to be recalled early. It happens, just as the weather could be too bad to a!low tendering so you may never get ashore.
If you do book private transport it needs to be cancellable, or you have cruise specific insurance.
Dublin Port is now too busy with freight to be able to handle too many cruise ship calls.

Thank you all for very helpful information! I rely on the RS Forum to assist when planning our travel itineraries, and these answers demonstrate why.

anne