I have two completely opposite solutions for you:
With a 3:00 a.m. flight, that means you need to be at the airport by around 1:00 a.m.
Solution 1:
- You leave your luggage at the hotel (just ask at the reception, they
will keep it for you)
- Around 9 p.m. you go to dinner in Athens.
- At 11:00 p.m. you return to the hotel to pick up your luggage and
take a taxi to the airport.
Solution 2:
- If you want to rest before the flight, you extend your stay by one
night in the hotel where you are staying.
- You go to bed (very) early, leave the hotel around midnight and take a taxi to the airport.
- You will eat something at the airport (not sure that everything will
be open)
My favorite is solution 1 :)))