Best Private Tour Guide for Athens

The Athens Half Day Tour which you mention is one of dozens of other transport services that will take you around in a comfortable car and stop from time to time in front of the indicated sites.
This does not include tickets for the good reason that they are not allowed to give you the guided tour of the sites.

For real guided tours (with entry to the sites) and complete and professional information
the only official tourist guides authorized to accompany you on visits to the various Greek sites are listed here:

https://www.tourist-guides.gr/

You will find the contact information, email, telephone of these guides.

If you do not want to go through a guide you can buy your tickets on the official website of the Hellenic Organization of Cultural Resources Development here:

https://etickets.tap.gr/webengines/tap_b2c/english/tap.exe?PM=P1

For Athens choose the region "Attica & Central Greece"

When we were researching a private guide for Athens we connected with Matt Barrett’s Greece website. We were interested in a comprehensive walking tour that included Jewish Greece as well as non-Jewish Athens. We hired the knowledgeable and engaging Salvador Levy. We have hired licensed guides throughout Europe & beyond. We put Salvador at the top of our list of favorite guides. If interested: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjn5eG18Oj7AhWpHzQIHUQpD5gQFnoECCAQAQ&url=https%3A%2F%2Fwww.totallyjewishtravel.com%2Fblog%2Foverview-jewish-athens&usg=AOvVaw272HERr9Df4tqRrOK1l8GL

YOu really really have to understand the difference between a Licensed Guide (who studies for years and takes very difficult tests & is trained); these people can take you inside of museums and ancient sites and churches and give you full information personally. For private parties of up to 3-4, their usual charge these days is about €60 per hour, w. 2-hour minimum. This is one of the websites https://www.toursbylocals.com/Greece-Tours where LICENSED experienced guides describe customized tours they can provide. MAny licensed guides are also teachers/professors, and the best are imbued with enthusiasm & passion for the places you'll see.

For a bit less, you might check the well-regarded Athens-Walks https://www.athens-walks.com which has a selection of tours with max 15 people & licensed guide. One tour does Acropolis in AM and Acropolis Museum after lunch... they also have a mid-afternoon Acropolis tour, when the big crowds will have left. NOTE: AthensWalks can also create a personalized private tour by LICENSED guide especially for your 4, to all the sites you want. NOTE: if you want a car tour with such a guide, you provide the car, or arrange with them for car, at extra fee. (Actually the important sites in Athens are all clustered closely within walking distance, many of them on "no-cars" lanes... so that cars can't get close anyway). NOTE #2: In all cases, the admission fees are in addition to the Guide services.... if they are included, it is spelled out.

What you should NOT do is fall for ads for "Day Tour Driver" which promises to drive you around to all the sites, take you to a wonderful secret lunch spot (PS its owned by his uncle), and tell you all kinds of "inside stories." You're essentially hiring a very expensive TAXI & a driver with the Gift of Gab. These people are NOT Licensed Guides -- they are not even allowed to accompany you into the museum or ancient site; they just drive you there (or several blocks away), tell you anecdotes & give you colorful folders that supposedly help you understand the exhibits or the ruins.

We hired Kostas owner of Athens - Walks. We had a fabulous tour with this local Athenian.