Acropolis morning or evening, early June

I think I would do the late afternoon/evening visit, in your circumstances. I do not know how late it is open, so be sure to give yourself enough time. I think doing it the morning and then having to get to the airport will have you watching the clock, rather than enjoying the experience.

My $0.02

We went to Acropolis in early June about 4:30 with a guide. I thought there were a fair number of people there compared to other places we had been in Greece. Our guide told us he had been there with another group at 8 am that morning and that it was far less crowded now.

During my last visit to Athens I went to the Acropolis to see the Parthenon when they first opened at 8am. It was a magical time with very few people milling around. There was also a military guard up there then for the daily ritual raising of the Greek flag, an extra bonus I was privileged to experience for the first time. Get up early and beat the crowds as well as the heat.

I'm with Joe, go later in the day if it's open late enough to give you the time you want up there. The crowd from your cruise ship will be just part of much bigger crowds. Head for the Acropolis Museum first if you have time after getting to Athens, see the sculptures that used to be on the hilltop. On Sunday morning you could visit the Agora, with its small museum, where you could spend as much or as little time as your schedule allows.