If you don't have a car then Killarney makes a good base for seeing the SW of Ireland. There are various tours you can get from there to the Ring of Kerry, Dingle, Gap of Dunloe, etc. Also, if you stay in town it is small and walkable and you will be able to get to shops and restaurants with ease. Definitely 2 in Killarney.

The problem I see here is the early arrival at Shannon (6 am). If you do not rent a car there at Shannon, the bus will take you to Limerick too early to check in at your lodgings.
You could transfer to a bus to Killarney in Limerick without an overnight stay. By the time you get to Killarney your lodgings would be closer to being ready for you to relax a bit.
Then if you decide to rent a car you can do that at the Kerry airport the next day or so. You can always visit Limerick on your way back to Shannon.
So with the early arrival in mind I would go directly to Killarney for the 3 nights, with or without a car unless you have rented lodgings for the night before your arrival in Limerick. Find out what the check-in time for the lodgings in Limerick will be. Many times in Ireland it is late afternoon. It is all a matter of timing to relax after an overnight flight.

We just returned from 2 weeks in Ireland with one whole week in Killarney. With you staying at the Ashville House you will most likely want to take a taxi from the bus station to the Ashville House because those two locations are on the opposite ends of the Killarney town center. I'm sure the B&B will let you drop off your luggage even if your room is not ready. Since you like walking you could walk into town that first day until your room is ready. The next two full days in Killarney you might want to do two day tours.

We did the Gap of Dunloe tour through Deros Tours (www.derostours.com) The bus from Killarney takes you to Kate Kearney's Cottage where you hire a jarvey with a pony and trap to take through the Gap of Dunloe, which is between two of the highest mountains in Ireland. Then you are dropped off at Lord Brandon's where you can have lunch. Then a boat ride will take you through the lakes and drop you off at Ross Castle where a bus will take you back to Killarney town center. You could probably arrange for the bus to pick you up at the Ashville House because it is right on the road that the bus will take to the Gap.  We also drove to Dingle on one of our days, but you could take a bus tour through Deros and also be picked up at your B&B. We have done the tour bus to Dingle on past trips to Killarney and that is a nice tour also. If I only had two days in Killarney those would be my two favorite ways to spend the time for two days without a car rental. Check the weather to choose the better day for the Gap if you decide to take that tour.