Italy usually shuts down for the month of August so a lot of the tourist attractions are closed. If you are looking for a mix of being by the sea and a bit of culture why not try Barcelona, It's a great city. Has the beach and lots else to do.
I have been in Rome in Early August and it was very hot. We had no problems with any of the sites closed. This was more of the case in Florence where they will head for the seaside.
We went to Florence 3 days Rome 4 days and Sorrento for 7 days.
We took the train between each.
With 4 days in Rome we did not see it all. If you want to go to the Vatican - give yourselves a day
Sorrento is about 50 mins south of Naples, and you can go to Pompeii, out to Capri lsland. You will not get much sandy beaches but take a look at the rest of the Amalfi Coast
you could also stay in Ostia which is easily accessible from Rome. This is the closest beach to Rome and very popular with locals. You could also visit the ancient ruins of Ostia Antica while there
How about a few days in Rome and then fly from Rome (Ciampino) to Barcelona (Girona) for the beach and fly back to Dublin from Barcelona (Girona)?
I visited both cities in one trip this way a few years ago, both are wonderful. You can get an apartment very close to the beach fairly reasonably in Barca, in the Barceloneta district (where there's a metro station, though it's not far to walk to Las Ramblas, El Born etc).