When: Anytime unless its raining or too hot. In summer time try to go as early as possible so you’re not in the full heat during midday.
Price: There is an entrance fee for the Parthenon as well as the Acropolis museum. But also some parts you can walk for free. If you are a student it is completely for free no matter which country you’re studying in.
Do: Definitely visit the Acropolis Museum. It is amazing and has also a cafe with view on the Acropolis. But the best part for me is the old city underneath the museum. You can still see the houses and shapes of the streets of ancient Athens. So far you couldn’t do a tour there but sometime in spring of 2015 they want to open it for the public. I’m very much looking forward for that on my next Athens trip.
My tip: Spend some time in the park below the Acropolis. There is a big stone you can get on and have a nice view over the city. Have a nice break there instead of going to the overpriced cafes of the area.