Ireland is a lived in landscape. Most of its upland areas can be walked across inside a day. I see it as a place for bike 'n' hike. Cycle quietish roads to the foot of a shapely mountain, pitch camp and then hike next day. Cycle on next morning. Macgillycuddy's Reeks have a fantastic ridge for anyone who enjoys a bit of a scramble. On the way west, you could take in the Galtees. Then head north for Mweelrea and Ben Bulbin. The roads are much smoother north of the border, where the Mournes offer great hiking. By the time you got back to the plane, at least three weeks would have gone on this itinerary and it omits islands, which would be a crime if making just the one visit to the Emerald Isle.
Consider the Pyrenees. They used to be much less crowded than the Alps (in the time of Franco). That's less true now (that Spaniards have money to spend) but they still have some wild bits. If it must be the Alps, the Vercors, just south of Grenoble would provide lovely backpacking, with bags of WWII history and some decent culture, for somewhere between 2 days and a week. I particularly enjoyed the Ferme de Lente Gite. Very friendly.