We arrive in Athens on morning of Oct. 9th. What is the best way to tour in 1 day using a tour company that would pick us up at the cruise ship port with our luggage and take us back to our hotel (Gate Hotel) after the tour?
You don't really need a tour company. If it doesn't include an official licensed guide, they aren't allowed to accompany you inside to visit the Acropolis (where the Parthenon is located).
The Gate Hotel is barely a 5-minute walk from the southeast entrance of the Acropolis. You will only need a taxi to get from the cruise port to the hotel, where you can drop off your bags and head right out to tour the Acropolis on your own immediately after. No honest taxi driver will drive you the 350 meters between the hotel and the Acropolis entrance, especially since the street layout makes a direct route impossible, meaning the drive would take 3 times longer than walking!
You can also visit the Acropolis Museum, which is on your way.
Itinerary:
https://maps.app.goo.gl/UZguoZMBKjzCWXLC7
There are free audio guides for the Acropolis available, including one you can download from Rick Steves' website.
You will need to pre-purchase your tickets several days in advance on the official website. This requires picking a specific day and time slot, and you must arrive at the entrance at your chosen time. All the morning time slots for October 9th are already sold out; choose a 3:00 PM or 4:00 PM slot instead, which will give you time to wander around the city and grab lunch first.
The only official website: https://tickets.hh.gr/en/venues/acropolis-of-athens-tickets
The official audio guides: https://www.hh.gr/en/digital-guide/