Thanks for your help. I just thought a boat might be an option.
The buses are cheap and reliable. If you're staying in Fira, which is the transportation hub of the island, it makes things very easy.
Akrotiri opens at 8.00 a.m. is and 7 days a week for the summer (usually closed on a Monday). You could take the first bus at 8.45 a.m. and arrive at 9.05 a.m. The bus will let you off right outside the diggings.
Depending on how long you spend inside (I spent three hours last May, others won't take as long), head back outside and get the next bus. One tip: The bus turnaround is one stop further down at Akrotiri Beach so you might have more chance getting a seat if you walk down there - its only a couple of hundred metres.
Once back at Fira, that still leaves plenty of time and multiple buses to get to Oia.
There was a water taxi starting from Akrotiri Beach that called at White Beach (pretty much inaccessible via land) and then Red Beach for €5 return. I don't know if they're still operating now that Red Beach should be avoided - its been subject to landslides and is unsafe to visit.