3 Days in Athens

I try to plan outdoor activities primarily on arrival day--that helps with jet lag. I would do the Acropolis tour though--it is outside and the major wow moment of Athens for me was the Acropolis, so I think I could handle that kind of tour.
I'd save museums for day 2.
Day trip day 3 if you want it (would depend on details). Enjoy!

I guess the way to deal with jet lag varies from person to person.

I would not visit the National Archaeological Museum on the first day. The museum is not in the city center in Plaka where the major sites and museums of Athens are and where you will probably stay, and it is huge. Huge enough to completely exhaust you if you are already a little tired and disoriented by the jet lag.

So, stay in Plaka, take a stroll in the streets to feel the atmosphere.

You can go to the Acropolis Museum which is nearby, or even to the Agoras and to Monastiraki Square. Everything is easily doable on foot on 90% pedestrian streets.

On day 2 do not expect to visit "all the historical sites in Athens" in one day. You would need at least a week.

Book tickets for the Acropolis early in the morning (booking is done online and you must choose a slot time).

Then you will have the choice among all these places:

https://www.google.com/maps/d/edit?mid=1OhGNGU1LUEgwFTul8u4YEe_tijF8Hqs&usp=sharing

What you want to do or see depends on your preferences, anyway you will have to make choices: Visit Athens or do day trips.

Also remember that it can be very hot in Athens in September, sometimes over 95F.

I fly from the US and usually arrive early to late morning.

Normally I head straight to my destination but there have been times when I couldn't make a connection so I stayed in Athens.

You'll be surprised how much adrenaline you have after a long flight and after I check in head out and walk around.

It may depend on your health, age, and condition you are in.

Most people recommend saving Athens for the end of your trip to avoid any delays or other factors that could cause you to miss your flight home.

The best time to visit the Acropolis is at first opening at 8 a.m. to eliminate the numerous tour groups and hordes of tourists that start showing up at 9 and gets progressively worse. Second best is late afternoon.

If there is a flight to Naxos upon your arrival and there is time to make it then most say to go directly to your destination.

Also recommended is at least 2 days in Athens with 3 better as there is more to Athens than just the tourist sites.

Sounds like you are interested archeological/historic and there's more than enough in the city to keep you busy plus there are areas of the city to check out.

Naxos has archeological sites, numerous Byzantine Churches, old villages and lots more..

From your advice. It sounds like it is best to spend the 3 days in Athens. Knowing this is a tremendous help! Trying to fit in too much is never a good idea.

Any recommendations for off the beaten path/non tourist spots to visit in Athens?

I did read saving Athens for the end of the trip was recommended however due to meeting friends in Naxos I had to schedule Athens first.

Looking forward to our first visit to Greece.

It's kind of hard finding something off the beaten tracks in Athens.

A better word would be less-touristy Athens but even then you'll find tourists but not as many.

Try the Central Markets on Athinas St. as it has meat, seafood and produce sections.

It's quite wild especially the meat section where you'll see things you won't back home. Nice seafood section but don't wear sandals and across the street is lots of produce vendors.

Consider Anafiotika on the north side of the Acropolis. It looks like a Cyclades village with white washed houses with colorful trim, very narrow lanes and views of Athens.

Kerameikos Cemetery is usually missed but it's a tranquil place in the midst of Athens and started out as ancient Athens pottery center but then converted to a cemetery. There's a nice museum with artifacts from the site.

The National Gardens behind Syntagma Square is quite nice and peaceful. Lots of plantings, small pond, walkways and you'll never guess you are in a city of millions.

There are several authentic Greek Neighborhoods like Thissio where you'll find more locals than tourists plus several others.

I'm sure someone will recommend other areas that don't see more tourists than locals.

As Tommy said, in 3 days you won't really have the opportunity to go anywhere other than the tourist spots.

Staying only in the historic center, the Central Market is an interesting place (the favorite destination of all food tours) and easy to reach on foot. Go there in the morning.

Still in the "market" spirit, you can also go for a walk to the Flea Market which is mostly a lot of souvenir shops but with a few second-hand dealers in the small perpendicular alleys.

The mini-neighborhood of Anafiotika can be explored for a short hour... provided you know where to enter (and exit) :))
I made this small map that can help:
https://www.google.com/maps/d/edit?mid=1ljBd_LqYpmCOpd5bVmAyMvR6P5cAbBA&usp=sharing

It also mentions the Flea Market

Crossing the National Garden there is an exit near the Panathenaic Stadium: (you can sprint there and be cheered by your friends, like an Olympic champion) : https://www.panathenaicstadium.gr/en/home/

In the post-industrial genre I also like the Gazi neighborhood, easy to reach by continuing west after Thiseio. You will find the Industrial Gas Museum right in the Technopolis park. Yes, you can also visit huge pipes in Athens!
The place is also a venue for events and concerts.

https://gasmuseum.gr/

If you go to the National Archaeological Museum, it is possible to go for a walk in Exarchia neighborhood which is just behind. It's the historically anarchic and protest neighborhood where you will find a lot of street art, music stores, book stores, and many cool bars full of young people.

There are many other interesting places little known by tourists to explore in Athens, but for you it will be for next time.