Even without golf I'd try to make it to St. Andrews, especially on a Sunday. The town itself is magnificent. Plus you can get a feel for the golf on Sunday by walking across the 1st and 18th fairways of the (closed) Old Course to the beach. Also the Old Course Hotel where the tee shot on 17 travels over the corner of the property itself.
Your route through Ireland is almost exactly the one I took in 1990 while attending Wimbledon and the British Open at St. Andrews. I was just a bit different in the southeast because I wanted to visit Tipperary town and also the home of Waterford crystal.
It's amazing how many people have asked me if I kissed the Blarney Stone. Fortunately the answer is yes. The first time as a teenager I thought it looked so stupid I was right there atop the castle but balked at doing it. Luckily I matured so when I visited on my own I made a point to rectify. They give you a certificate of accomplishment. Plus the grounds are extensive and a good tour on their own right.
To the northwest near Galway I love the rugged Connemara section, including the narrow Sky Road. That was so spectacular I made sure to do it again during a return visit.
Pick out a few castles to visit. There are so many I hate to give recommendations. I enjoyed every one.