Dick’s suggestion was my first thought, too. End your tour early, in Galway (which is worth at least an extra day or two more than your tour is probably spending there), then get to the Aran Islands from there. On 2 trips to Ireland, both included time in Galway, and trips to Inis mor, the biggest of the 3 islands - definitely not to be missed.
First trip, we took the ferry there. Rough seas postponed getting back to the mainland, and we wound up on a ferry that stopped at Inis Oírr, the smallest, before getting to the mainland with yet another torrential downpour that day. Second trip, we flew. It’s a very short flight, and planes go when ferries are stuck at port. Our second trip, with a severe storm on Day Two (after sublime Day One weather), we flew out in the afternoon when the weather cleared a bit, and the ferries were still no sailing. Definitely pay more and fly - it’s worth the time, money, safety, and keeping your schedule going.
Our second trip, after Galway and the big Aran Island, we drove north, eventually reaching the Antrim Coast, ten curving down to Dublin to turn in the car.. We were able to rent in Galway and turn in Dublin - all in the Republic, even after driving into Northern Ireland for some time. Renting in Northern Ireland and returning in the Republic (or vice versa) is probably impossible.
As for worthwhile sights between Galway and Portrush, those include (and you could do these in reverse):
1) Hawk Walk at Ashford Castle, outside of Cong. Rick Steves recently started including this in his guidebook, saying he didn’t know that it was on his bucket list until after he’d done it. With a guide, you go on a walk in the woods with your own private Harris Hawk (can also be shared between two people), and a big glove, and you spend a fascinating time with your bird, it flying from you and back to you, as the guide teaches you about the ancient art of falconry. Go for the Extended Walk, which includes time with their cool owl.
2) Killary Fjord Boat Tour at Leenane. Ireland has one true fjord, and unless you’re also going to Norway on this trip, this is your one chance to cruise up a fjord. It’s stunningly beautiful, and the boat trip up and down passes lots of mussel cages - they are farmed in the clear waters. We hadn’t had lunch yet, so ordered mussels onboard, and enjoyed them as we sailed in the waters from where they’d come.
3) Achill Island, which is connected to the mainland by a bridge, but it’s another island (like the Arans) that should be seen to be appreciated. We spend a night at a B&B, but at the very least, spend part of a day. There’s a golden beach at the west end, and haunting ghost town remains from the Potato Famine abandonment.
4) Sligo town - then we actually stayed 2 nights at a B&B in the country a few miles northwest of the city.
5) Slieve League cliffs - north of Donnegal, these are even taller than the Cliffs of Moher south of Galway, which I’m guessing your bus tour will include.
6) We actually stayed in Bushmills, just a bit east of Portrush. It was a better base for hiking along the Antrim Coast to the Giant’s Causeway. The town also has the famous whiskey distillery, and our final day, we stopped off there for a tour, but were told a large family in a motorhome had just arrived ahead of us, and were getting a private tour, so we’d have to wait for the tour more than an hour later. We didn’t stick around to be allowed on the later tour, and continued on towards Belfast.
