No, there are trains to transport you from many major towns to other major towns. There are taxis available while you are in town.
If you plan to see Dublin or other counties up north I would fly into Dublin or at least plan on flying out of Dublin. The closest airport to Killarney is Kerry, it may make the most sense to fly into Dublin then change flights to fly into Kerry. Especially if you don’t plan on renting a car.
It depends on what your plan is. If you plan to stay in Killarney for only the weekend I would look at a hotel or B&B. If you plan to stay for a week I would look at the cottages.