Dublin Airport is sadly lacking in shower facilities and places to relax. I'm sure you could go to a hotel, but they will probably charge you a full day's rate. Best thing to do is contact a few and see what they say. If you go to Google and type in "Dublin airport" + hotel, it will bring up all the hotels in the vicinity of the airport.
I know the Great Southern used to offer rooms for a few hours but it has been taken over by the radisson so not sure if it still offers the service. You can try there or the Clarion which is the other airport hotel