With four nights to work with, I would opt to leave out the south of the country and stick to the north and west of the island. Otherwise you'll just end up having to ride through everywhere without really being able to see, stop and enjoy the places you visit. Better to see lots of the little you visit than little of the lots you ride right through.
Even when opting for A roads, the average speed you'll achieve can be slower than expected. This will be even more evident when you pull over to check out things of interest along your route.
I would advise going as far south as Galway or maybe Limerick. Both places offer a pretty quick return route to Dublin once you've covered as much of the WAW as you can fit in.
I've not ridden around Donegal so can't honestly say if it's worth including, but once you start heading south I would recommend the waterfall at Glencar, the N16 out from Sligo town is a lovely bit of road with great scenery too.
Strandhill is a nice spot to grab a coffee and an ice cream beside the beach - especially if the weather's nice (sometimes it does happen!).
I would second a visit to Achill island - one of my favourite spots along the west coast. Make sure to ride all the way out to Keem beach - you won't be disappointed. If time permits, ride up to the summit of Minnaun, on a clear day the view is spectacular. You'll probably use up a whole day doing Achill, but it'll be a day well spent.
From Westport head for Leenane on either the R335 or the N59 - neither will disappoint but the latter is a more direct route. Continue onto Maam cross and Oughterard, tis a lovely bit of road. At this stage you're only a stones throw from Galway. If you opt to overnight here, I would suggest staying in the Salthill area. Tis a bit quieter than the centre of Galway, and it's a handy spin into the city centre in a taxi if you opt to head in.
If time is on your side, head south from Galway on the N67 towards Kinvarra and Ballyvaughan. A little after Kinvarra detour onto the L1014 to take in the Hazel Mountain Chocolate shop and cafe. Nice spot to stop off and really indulge! A little further on, turn right towards Bealaclugga to get back onto the N67. Just after Ballyvaughan take the R480 up to see the Aillwee caves and the Poulnabrone Dolmen. Yer not far from Fr. Ted's house, and the small back roads are a nice gentle spin with some crackin' views of the Burren along the way.
An overnight some where in Co. Clare like Doolin or Lahinch should find you nicely set up in a local pub with some decent trad music being played. Also, gives you a chance to visit the Cliffs of Moher if you haven't been before.
Fastest way back to Dublin is via the M7 from Limerick, not the most interesting ride, but it gives you more time to spend along the picturesque west coast.
There are many options of places to see, roads to ride etc. - way more than a four night trip can cover. But the above route catches a couple of nice spots along the way, with plenty of things to stop at and take a look. Hopefully a few others will have more to add which gives you plenty of fuel for thought.
Let us know it you want more info - places to stay etc. I can rattle on about this stuff till even my mates have left the room in desperation!!