How many days are needed to tour Athens well?

It all depends on what interests you. For the major archaeological sites alone, you'll need two full days Most of them are located in the same largely pedestrianized area.

Add a day or two to see museums, many of which are also located in the same area.

If you're tired of ruins and museums and want to see the city in a different way than rushing from one site/museum to another, add another day or two.

So, to find a compromise between the time you have and enjoying Athens, let's say 3 or 4 days is a good amount.

By the way, there's no point in taking the Hop-On Off buses; you won't see anything except the busy, polluted main avenues of Athens.

The official Athens website:

https://www.thisisathens.org/

I don’t have an answer to your actual question. If you are going to spend some time in Athens before the cruise, you might want to use your day of cruise excursions to see something outside of the city .

If your cruise includes one day, I would suggest at least 2 additional days, or more depending what the cruise tour includes. You want to see the Acropolis, Acropolis Museum, Archaeological Museum, Mt Lycabettus, Agora, Forum, etc. We also did a food tour, which was great. You won't run out of things to do!

hey hey Lydie
when is this cruise, what cruise line and where does the cruise end at, how many days? if it's me i would go 3 or 4 days before to see sites without being rushed. there is much to do & see. what are you wanting to see and do? like the idea of doing a food tour and even a market tour.
if your have a tour in athens with ship, go somewhere else as another poster mentions.
about 10 years ago we did a R/T greek island cruise from venice to venice. it was great on NCL. if your end is venice which is an irritant to me, no stops there, couple other ports that needs transportation to venice by train, bus or taxi ride which could be couple hours. have fun and enjoy
aloha