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.

I was not overly impressed by the Acropolis Museum. However, it is worth going to the top-floor restaurant (free access) for the rooftop view. You don't have to order anything but it's a very good place for a drink or a meal. You can see some of the copies of the Elgin marbles in the museum through the plate glass windows separating the museum from the restaurant/gift shop.

Chani, thanks for the tip. I’ve read elsewhere that it isn’t worth buying tickets to Roman Forum, Hadrian’s Library and Aristotle’s School. Also saw that Panathenaic Stadium tickets are only sold at the sight.

Your list looks really good. Athens is very compact and you can see a lot in a short time. Walking everywhere saved us a lot of time except when going all the way across town. We managed to tour many of the neighborhoods on our way to various sites.

When we were in Athens last October, we covered several of these the day we arrived and did not need tickets. We like to get out and walk on arrival day to keep the jet lag at bay. We were staying in the Mets neighborhood and walked to Aristotle's school and looked at it through the fence along the road. You can then go over two streets and walk past the Presidential Palace. Continue on south along the street in front of the Palace and cut through the Zappeion Gardens to Hadrians Arch. The trees in the park are filled with nesting Parakeets. The Temple of Olympian Zeus is a little farther around the park and can be viewed through the fence. It has scaffolding covering it so there isn't much to see. We took our time and this took a couple hours at most.

The Ancient Agora is quite large, so give yourself time to explore. There is a wonderful restaurant right outside the exit with lots of outside tables looking straight at the Acropolis. The Moussaka there is awesome! The National Archaeological Museum also deserves some time and is very worth it. These get crowded, but we did not wait long in line for tickets.

A couple of things I would recommend that are not on your list are the spice markets of Evripidou Street and the funicular up Lycabettus Hill. Happy travels!

Thank you, travel scamp, for your overview. Fingers crossed we land in mid May. With you on walking off jet lag. I'm adding your terrific suggestions to my list. Parakeets in Athens? You scooped the travel guides! They list so many souvlaki stands, cafes and restaurants, my head is spinning.

The Ancient Agora and the Nat'l Arch. Museum were my favorites. I looked at your #5, 6, and 7 from the street and that was fine. If you are there on a Sunday, try to fit in the changing of the guard at the top of Syntagma Square and the Tomb of the Unknown Soldier. I believe it begins around 10.45 but best to be there a few minutes early to get a good spot. You'll hear the marching band as the troops approach. There's a changing of the guard every hour on the hour 24/7 but only once a week is there an elaborate ceremony with a band.

Thanks for the tip, Chani.