Athens for a day..What to do?

Visit Acropolis and the Agora below it. If you have enough time, the Acropolis Musuem is so worthwhile. I avoid the Plaka as it is filled with touristic shops and restaurants. Near the Acropolis Museum is a good restaurant, ManiMani. Monastraki would be good too.

With your plane arriving noonish, it is unlikely you will have time for anything more than you mention.

If you arrive from another Schengen country, you do not have to do passport control which will help cut your time at the airport. If arriving from a non-Schengen country you do have the delay of passport control.

The time to get to Athens and presumably check into a hotel, means you will likely be arriving at the Acropolis not much earlier than 230. If time permits do see the Acropolis Museum.

My favorite restaurant is on the street on the "back" side of the museum

https://www.tripadvisor.com/Restaurant_Review-g189400-d2108580-Reviews-To_Kati_Allo-Athens_Attica.html

Thank you for the restaurant suggestion. The museum is added to the plan too.
Where (neighborhood) should we have dinner and walk around that night?
If I had an extra day in Athens, should I stay in the city to see more or visit the island of Hydra for the day?

I agree with Joe that the Kati Alo is nice. I've been a couple of times this year and enjoyed both times.

Another option, only five minutes from the museum, is the Byzantino. We've written up our take on it here.

http://www.greekramblings.org.uk/Plaka_Monistiraki_Restaurants.html

Assuming you are there in the summer I would go for the Byzantino over the Kati Alo as it has a far bigger outside.

Alan

Two suggestions:

(1) Anafiotika on the north side of the Acropolis. Since it's close to the Acropolis it would be easy to add to your schedule. It's a small area with white-washed houses, narrow lanes and lovely views of Athens. It's like being in a village on a Greek Island and you'll never realize you are in a city of millions.

(2) Another option just a bit further away from the Acropolis and often overlooked by tourists is the Kerimikos Cemetery which was the ancient burial ground for Athenians and others. Lovely area, quiet, peaceful and filled with archeological remains. Very nice museum with artifacts from the site.

I'll add a third suggestion: If you really want a wild, crazy and great experience check out the Central Markets on Athinas St. . . . it's where locals and tavernas owners go for every day shopping and is filled with sites, sounds, aromas and seafood/meat/produce that you may not have back home.

Athens has lots to offer, just not the well-known tourists sites but non-touristy and every-day areas of the city.

Wow. Great suggestions. Thank you.