You're right the sites will be closed, and there is little public transport on the 25th.
A good bet, assuming the weather is OK, is a walk on Filopappou, the hill opposite the Acropolis. There is a small monument on the top and great views back across to the Parthenon. We've written up some directions. The second walk down on this page
http://www.greekramblings.org.uk/Exploring_Athens.html
The whole pedestrianised walk around the Acropolis - along Robertou Galli and Apostolos Pavlos - is worth doing. It's also worth noting that while you can't get into the Acropolis you can get onto the Aeropagos just below it.
In terms of eating pretty much everywhere outside Plaka, the central tourist area, will be closed but most places in Plaka are open. We sometimes spend Christmas night in the Byzantino which generally has a lively atmosphere. It also has an outside area with very effective space heaters. Again we've written it up
http://www.greekramblings.org.uk/Plaka_Monistiraki_Restaurants.html
Have a great trip. We really like Athens at that time of year, even though places are closed.
Cheers
Alan