There are two places I went to that are not Camino-related, but are very much places of pilgrimage for the Irish, and both are very beautiful.
The first place was Glenadalough, where St Kevin lived, and the site of a monastery from the sixth century. You can get there by daily bus from Dublin, and there is a youth hostel to stay in. It does get busy with tourists around the village, checking out the extensive monastery ruins, but there is a superb walk, where you soon lose the crowds. I went on the Spinc and Glenealo Valley walk, which heads up high and gives superb views if you are lucky enough to get the good weather I got.
The other place might be harder to get to on a short stay, as you need to get over to Westport in Co Mayo, but it is the very famous Croagh Patrick, place of pilgrimage to St Patrick. This is a much more strenuous walk uphill, especially the last third of it, where it is both very steep and very rocky underfoot. Conditions can also be bad weatherwise here, and they don't recommend climbing it in the rain. (It has many loose, slippery rocks.) I struck very bad weather on the descent even in August, but it was an amazing walk to do.
Margaret