There's a lot of information on both Fodor's and Frommer's sites.

Bunratty Castle is near the airport; maybe save it for your last day. With only 5 days you may want to stick in one general section of the country, and, if you want to go to Cork, you may want to go south from the airport, travel through Kerry and on through Cork and Cashel and back to SNN. The Cliffs of Moher are east of SNN in County Clare. Galway is a bit north. Belfast seems too far.

Here's a hotel guide - once you get your geographic bearings, look into the B&B and pub accommodations. You'll save by staying in small towns. In the off season, many places can be found for less than 50 Euros/night.

http://www.irelandhotels.com/