If you only have 3 or 4 days to spare, then you'll want to make them 'quality' days, so please don't go to any of the places that have known access problems. Hmmm... that pretty much rules out nearly all of the West of Ireland! No... there are still places you can walk... where local folk are actually pleased to see you.
Try some of the following, which are all OK...
1. The North Mayo Coast - hilly all the way and fabulous sea views.
2. Achill Island - especially Croaghaun and Achill Head.
3. The Bangor Trail - a really tough little mountain/bog trail... or...
4. Nephin Beg Range - using the Bangor Trail for access.
5. Croagh Patrick - yeah, it's very popular, but still a fine mountain.
6. Mweelrea - but only if you can get on and off it unobserved.
7. Maum Turk Mountains - and end-to-end, if you can arrange a lift.
8. Twelve Bens - but awkward if you want to park a car in some parts.
9. Brandon - excellent mountain if you get a fine day with good views.
10. Carrauntoohil - via the Coomloughra Horseshoe if possible.
11. MacGillycuddy's Reeks - end-to-end, with or without Carrauntoohil.
12. Mangerton - but only if it's a clear day, so you can get the views.
Most of those are in the book, and the North Mayo Coast is pretty obvious anyway. Access generally in Ireland is just about the worst in Europe and it certainly isn't getting better. Some years ago, when insurers and politicians started putting the frighteners on the farmers and landowners, it pretty much meant that asking permission to cross someone's land was invariably met with a blunt refusal. The only other option was blatant trespassing, or entering into hefty negotiations and producing evidence of insurance cover and an indemnity for the landowner in the event of any accident. It's a complete mess and no-one's doing much to sort it out. A farmer put it rather more simply for me and said - "You can walk wherever you like, but for God's sake don't ever ask for permission, as the answer will always be NO!" Another cunning ploy was for farmers who hated their neighbours to gleefully point out routes over their neighbours' property!