Budapest Trip Suggestions

Look at New York Times 36 hours in Budapest.

The NY Times has lots of 36 hours in [European city] articles and they would probably suit your style of fast travel.

Nepliget is a bit "out there" but there is a metro stop near; still I would just call a taxi (CityTaxi --- google them and down load their app).
For one night the stay on the river; the Intercontinental or the Marriott.
To get to the airport call (or use the app) City Taxi the night before .... pickup 3 hours before flight is very safe, 2.5 hours will work too.
Go to google maps and look at the route of the 47/49 tram and the 2 Tram and the M1 metro; as they will take you to everything you have time for and are easy to access from a river front hotel.

EDIT (better Idea)
Hotel to Great Market Hall (walk down Vaci utca) to Great Synagogue (tram) to Andrassy ut (walk) to Hero's Square (M1 and walk upper half of Andrasy ut) to Basilica (M1 and walk lower half of Andrassy ut) to Parliament (walk) to Hotel (tram).

The only thing I might add as a relatively new traveler to Budapest is that if you don’t have the budget for the Intercontinental or Marriott, you can still easily access the metro or bus to the riverfront from your hotel for the evening lights. (I interpreted that you are staying at the Exe Budapest Center?) . Walk down to the Elizabeth Bridge and then ride the 2 tram up to the Parliament at night with the lights - it’s pretty magical. Then ride back to wherever takes your fancy. Stop for dinner somewhere on the river.

If I have your location correct, you will also be across the street from Mr. E’s current favorite pastry place Auguszt Cukrászda, and a 5 min walk from the Doheny Synagogue. If I don’t, go to the river anyway and ride the tram at night. Mr. E has given you a good day of sights for your next morning.

If you have data, I found the BudapestGo app easy to use. Be sure to scan the QR code each time you board. If you have the paper pass, don’t lose it. Lol.

I only suggested the riverfront for accommodations because there is something to see or do 24/7 and with 1 night I thought it best to put him/her/them/Z in the middle of everything; generally with more time I always suggest a little further out.

Those hotels are about $220, this one is half the price: https://budapest.zenithoteles.com/en/photo-gallery-hotel-center-budapest/ in about the same location.

But the Exe Budapest Center works pretty good too; about a 10 minute walk to the Corso ... through some nice areas.

Okay, TravelMom, tell me I have done it wrong all this time; but with a multi-day pass I never scanned anything, I just showed the screen when asked.

Is there an area that isn’t nice to walk through? I might have to walk a loooong way to find it…..

Me tell you you are doing something wrong? Not a chance and you should know better! Ha!
In fact, in re-reading, it still looks like you have to scan for metro, but not for trams and busses where you enter from all doors, if you have a 15 day or shorter pass. And I thought it was everything…. Oops.
https://bkk.hu/en/tickets-and-passes/mobile-ticket/
https://bkk.hu/en/tickets-and-passes/budapestgo/
https://bkk.hu/en/tickets-and-passes/prices/monthly-passes/15-day-budapest-pass.55/

But I would be willing to bet a glass of wine that you get the month long pass - and apparently showing it is all you have to do. And scan the inspector’s code if she asks you to.
https://bkk.hu/en/tickets-and-passes/prices/monthly-passes/monthly-budapest-pass.58/

I think I couldn’t sleep one night in Poland….. 🤦🏻‍♀️

One day and places to visit: The market and ride the funicular to Castle Hill. Both are in the same area of town and doable in less than a day. In fact you could probably do both in 6 hours giving you plenty of time to get to airport.

The market is a great place for a meal . Castle hill will allow you to experience the city in your short time frame.

Near the Synagogue are fun restaurants and bars for evening. You could do this area night you arrive.