I'm a long-time reader / lurker and frequent traveller, but rarely post since I don't have anything unique to add. I just finished a successful public transit itinerary to visit Orkney's main sights during a weekend break, basing in Kirkwall. This took quite a bit of trial and error, and only seems to work exactly one way. I thought I'd share in case anyone is planning a similar trip. In the end, I saw everything I wanted to see, with ease, despite limited Skara Brae buses and few Sunday services.
Background: I'm an infrequent driver (have always lived in cities, never owned a car), and I've never driven on the left. Although I read some advice that driving in Orkney is easy, I'm glad I didn't risk it. People are speed demons, and there are many areas with narrow roads.
Important bits: Schedules are as of June 2023. Make the reservation at Maes Howe as early as you possibly can, because this itinerary won't work otherwise. Arrive on Friday evening to ensure you get the Saturday morning bus, and leave on Monday morning so you have the full day on Sunday. Because of the bus 8S schedule, this itinerary will only work on Saturdays, not on other days.
Itinerary for Saturday:
8:30 Bus 8S from Kirkwall Travel Centre to Skara Brae, arriving 9:22.
You'll get there a few minutes before it opens, and most importantly before any tour groups arrive. I was nervous about limiting myself to 90 minutes at Skara Brae, but it was more than enough time for Skara Brae and Skaill House. There were < 10 people at Skara Brae with me, but then busloads of people were entering just as I was leaving to check out Skaill House.
11:05 Bus 8S from Skara Brae to Ring of Brodgar, arriving 11:39.
Get off and explore Ring of Brodgar, walk down to the Standing Stones of Stennes (approx. 20 minute walk), explore the Barnhouse settlement too. Then walk to the Maes Howe visitor centre (also approx. 20 minutes). All in all, you have approx. 90 minutes to explore both sites net of walking time. I found this more than enough and got to the Maes Howe visitor centre early.
14:00 Maes Howe reservation, tour will last just under an hour, including transportation back to the visitor centre.
15:13 Bus X1 from Stennes Garage (across from visitor centre) to Italian Chapel, dep. 15:13 arriv. 15:56
I arrived and had the Italian chapel all to myself, plus time to take a look at the Churchill Barriers. As I was leaving two enormous tour groups showed up, including the John O'Groats mega day trip.
16:35 Bus X1 from Italian Chapel to Kirkwall Travel Center, arriv. 16:52
Then on Sunday:
10:00 - Bishop's and Earl's Palace (they open at 9:30 if you want extra time). St Magnus will be closed for mass, but you will get to that at the end of the day.
11:26 Bus X1 from Kirkwall Travel Centre to Stromness, arriv. 11:56
I managed to do this walk https://www.orkney.com/things/leisure/walking/stromness-warebeth, climb to the Stromness Viewpoint, and check out the Stromness Museum all between 12 and 4. Skip the museum if you're not a fast walker, or just do part of the walk if the museum is more important or the weather is bad.
16:05 Bus X1 from Stromness to Kirkwall Travel Centre, arriv. 16:35
Then I swung by St. Magnus Cathedral, which opens from 13-17 on Sunday.
Due to the limited Sunday schedules, the pieces of the puzzle only work this way - don't try to swap anything!
I hope this helps someone out there, as it was the ultimate brainteaser for me.