Public restrooms can be scarce in Ireland. Just pop into hotels to use those in the lobby. The staff won't care if visitors come in off the street to use them.

Also, very few Irish public restrooms use paper towels. The country has almost completely converted to hand dryers. If you dislike waiting the time it takes to blow dry your hands, bring a small pack towel (available from camping stores) that you can put into your pocket or purse. It has the advantage of being environmentally friendly *and* fast.