Where to stay in Athens

To make the best use of your very limited time, choose a hotel near the Acropolis Museum, that end of town, like the Divani.
Prioritize the Parthenon with Acropolis, the Acropolis Museum. For a small and unique to Greece museum, the small Museum of Cycladic Art is special.
Take a taxi up the Lycabettus Hill to the tram that takes you to the top. Here you will see all of Athens spread out before you to the sea!

1.5 days is rather short to visit Athens, so to optimize your stay stay in the Plaka / Monastiraki / Syntagma Square / North Koukaki area.

From there you will be within walking distance of the main places to see and you can visit everything on your own.

In these areas you will find many hotels in all price ranges.

Note that a Hop-on Hop-Off bus type tour is totally unnecessary, and most private tours will just drive you from one place to another. Only official guides can accompany you to the archaeological sites.

For the archaeological sites you can buy the tickets on site or buy them on the official website:

https://etickets.tap.gr/webengines/tap_b2c/english/tap.exe?PM=P1

(Choose Attica & Central Greece region)

The 30€ combined ticket "Acropolis & Slope" gives you access to the main sites: Acropolis, Hadrian's Library Ancient Agora, Kerameikos Aristotle's School Roman Agora, Olympieion temple.
This combo is valid for 5 days.

The Acropolis Museum has a separate ticket office:

https://www.theacropolismuseum.gr/en

Other places like the Parliament where the changing of the guard takes place and the National Garden are also within walking distance.

There is no particular order to visit the sites except that which could be imposed by the climate.
It can be very hot from June, in this case avoid visiting the Acropolis between 11:00 a.m. and 4:00 p.m.

From the moment you have booked on the official website you automatically have a "skip the line".
Third party websites promoting this option do not provide anything more.

All of these basic activities plus city walks from site to site will keep you busy for 1.5 days.

Great advice above. To be a little more succinct, I would advise to find a place in the areas recommended and then immediately go to the Temple of Olympic Zeus. Buy your multi-site ticket there as there won't be a line at the box office and it's a worthwhile site on its own anyway.
I like to time my arrivals at the Acropolis for about 18:30 since it closes at 20:00. Far fewer visitors at that hour, a tiny bit cooler, and pleasing golden hour light.
Similarly, I like to arrive at the Agora first thing in the morning.
There are plenty of good hotel options for a Triple Room in all of those central areas for between $150 - $250 / night. That would include breakfast and there are other budget options.

We stayed in Athens this fall prior to/with an OAT tour, at the Athens Cypria. It was mid-October, so maybe off season, but rates were very reasonable. They may have room big enough for a family, or connecting rooms. Also, fabulous view of the Acropolis from the rooftop bar - do not miss this, especially at night! Very centrally located - we could walk to the hotel from subway station at Syntagma, and walked to almost all the central sites - Acropolis, Acropolis Museum, Agora, Hadrian's Library, The Plaka, etc.

Athens is super easy to get around, at least in our opinion, but sites might be busier in June? We bought a combo ticket near the Agora that included a bunch of the archaeological sites (but NOT the Acropolis Museum). We used the Rick Steves audio tour for the Agora (good), and for the Acropolis (not as good, but may have been because it was crowded, so had to stop/start audio frequently).

With such a short time frame, I would suggest you prioritize the Agora (Athens Agora, not the Roman one), Hadrian's Library, Acropolis and Acropolis Museum. Also, find a restaurant serving fresh grilled fish!

editing to add: if you have youngsters (or even older) Hadrian's library has the awesome addition of a tortoise...herd? that rumbles around the site

InnAthens hotel is near Syntagma Square. Easy walking distance to all the sights; modern hotel; excellent service. Suites are less than 300 euros per night. We had a terrific stay there.

Another affordable hotel is Attalos. Its near Monastraki metro station so close to most of the central all the main sites. Many of the rooms have acropolis views and the roof bar has an outstanding view of the Acropolis.
They also have family rooms with three beds.

I know this is a long time since your post, but we stayed in an NSplace apart in Athens. Right in the middle of everything. They have 1 and 2 bedroom apartments and it was very nice. Just look it up. We got it through booking.com.

We paid about 100eu for a nice hotel with street view, and buffet breakfast overlooking a city plaza. We were able to walk to the Acropolis about 25 minutes away. One friend paid about 69eu but did not have a window view of the street. It was also just steps from a subway station, and there was a 24 hour coffee and pastry shop next door (which came in handy when we left at 5 am). There was also a nice seafood restaurant where we paid 7 eu for a sea bass dinner.

Athens Tiare by Mage Hotels
2 Pireos Street, Athens, Attiki, 10431 Greece