As Sweeny says Athens is pretty safe no matter how you travel including on foot, especially during the day, so don't worry. As others have said you can take the new Metro from the airport and one of the stops is practically at the foot of the Acropolis. There are archaeological exhibits in the Metro in that neighbourhood.
You could go up the Acropolis and see the Parthenon (it's a much better temperature for that in Feb than in July when we went!) and/or walk around Plaka, shop, relax in a cafe or taverna...many of them have wonderful views of the ancient monuments. There are quite a few pedestrian-only streets in Plaka now, which makes it pleasant to explore.
Enjoy your day!