Andy, looks as if you have budget in mind; however, most budget places are small, family-run, & don't have vehicles for transfer. One that does is Anemomilos Suites - https://www.santorini.com/hotels/anemomilos-suites/ - this website specifically says that airport mini-bus transfer is complimentary. It also has a pool which will be welcome if your June day is HOT. I agree with your strategy, exactly what I did on my first visit. Also, my FAVE budget gem, Villa Fotini, is on the northern fringe of FIRA, and on the inland side, only 20 yards to the handy bus stop to OIA; also use of pool at a nearby hotel. Website says airport transport on request, but I don't know what the fee is. I do know that transfer to/from seaport is complimentary. Alas, Fotini now charges for breakfast ... but the location is super, and so is the family that runs it.
I'm not sure you're going to find a half-day tour, they're not as common lately (full-day = higher profit), but you can do considerable sightseeing on your own, if you strategize. (You'd miss the Akrotiri Dig). On arrival afternoon, I would advise to explore Fira town, if you wish, walk down to Old Port on steps (watch out for donkey-doodoo), and then zip back UP on cable-car. Stop by STANI taverna (on 2nd lane inland, north of Koo club), go up to roof terrace to check out its FAB view of caldera... then reserve a table for 7:30 or so. Go have a swim then back to Stani for sunset without crowds, noise etc etc. Authentic Greek cuisine, and more than 2 of you can eat, plus 1/2 Liter house wine, for under €30. In the AM, jump up early, and catch the FIRST bus toOIA ... 8 or 8:30 ... then y ou'll have about 90 minutes to wander its lanes, take amazing photos before the Deluge -- the arrival of the Dread Cruiser Invasion! Thousands and thousands of passengers, with lapel labels and dazed looks, cramming the OIA lanes. When u see them coming, time to go! Take bus back to Fira town, and enjoy the super Prehistoric Fira museum, showing all artifacts from Minoan culture of 5,000 years ago! You can enjoy a cheap & tasty gyro in the jammed FIRA town square, before heading out ... If you take that BLUE STAR ferry, run up to the top deck to get the departure view of the cliffs.