Can't believe no one answered this - the answer is - GO A DAY OR TWO EARLY!. If the ship leaves at 6:00, they should allow you to board up until then, but don't risk being late.
Actually, you can see the best of Athens in a few hours, it is called the Acropolis. Surrounding it is the Plaka shopping district, and the Agora. One full day is enough to see Athens, but with your schedule, I would just go to the Acropolis and that is it.
The problem is traffic can be terrible and by the time you get your luggage and make plans to get to the port, you won't have much time.
Also - taxis in Athens are tricky. They may not stop, are known to overcharge, and might go "the long way".
If the cruise lines offers an excursion, I recommend that. Otherwise, play it by ear. Get in a cab, go to the Acropolis, and plan to be in a taxi headed for the ship at least 90 minutes before departure (assuming the ship is at the usual terminal).
I am the editor, but I also speculate, ask questions and play devil's advocate. I reserve the right to change my mind.