Athens

Since I assume you're going to Athens in the coming weeks, it will be during the off-season. So you don't need to pre-book your tickets, although you could pre-book the Acropolis ticket, which requires specifying a time and date.

For all other sites, simply go to the ticket office.

Some sites can be visited quickly, like the Museum of Modern Culture, while others, like the National Archaeological Museum, can take up to 3 hours.
Some sites can be viewed from the outside without necessarily needing to enter and pay a ticket to get within a few meters.

In addition to the Acropolis , I would suggest going to the Acropolis museum and the Ancient Agora. The Ancient Agora also has a small museum on the grounds which is included with the entrance ticket. Your list does not include the Temple of Hephaestus which is located on the grounds of the Ancient Agora and is also covered by the entrance fee to the Agora. We went to Athens this past May with our son and we went home 2 days earlier than he did. He visited the National Archaelogical Museum after we left and liked it better than the Acropolis museum. When we went in May ,one of the days we were there was free museum day. We entered the Temple of Olympian Zeus for free . If we had to pay an entrance fee, it would've been 20 euros a person which definitely is not worth it as most of the building is covered in scaffolding.Other sites we visited that I would cast a negative vote against was the Roman Agora and Hadrians Library. It's a shame the multi-site combo pass was discontinued as that was a great deal.
As Jo Lui posted , some of the sites can be seen without entering the grounds and paying an entrance fee.

Thank you, JoLui & darkmist1, for the helpful info! Heading to Athens in May. Planning wayyyyy ahead.

For May, you won't be able to book anything before early April. The prices for sites and museums switch to summer rates on April 1st. So don't panic.

That said, plan to book the Acropolis at least 10 days in advance if you want to go for an early morning slot before 10:00 am to avoid the crowds of cruise ship passengers and organized tour groups.

I assume you already know this, but as a reminder, the Acropolis Museum is privately managed and has its own website and online ticketing system.

https://www.theacropolismuseum.gr/en

Since you're going in May and if you're not already locked in to your dates, consider including May 18th as one of your days in Athens.
That day is Museum day and most of the sites have free admission and that is what I was referring to in my previous post.
When we went last May, we saw the Ancient and Roman Agoras, Temple of Hephaestus, Hadrian's Library and the Temple of Olympian Zeus on that day and we walked right in. However, a friend that was with us tried to get into the Acropolis that day and at 3 pm it was already "sold out".
It is suggested that if you try to go to the Acropolis on Museum day that you line up by 7:30 am.