Hi, Judy B. Hotels are very individual, but you might find this post as a great place to start researching for what you want. Many of the ones listed are near Andrassy, which is probably where you would aim for.
https://community.ricksteves.com/travel-forum/hungary/rs-forum-favorite-hotels-and-their-cost
I personally have only stayed at the K+K, but others on the forum have stayed at the Casati, Moments, and Medos (I have reservations there for next year).
I would suggest 4 nights, if possible. You may arrive with jet lag and 4 nights will give you 3 days.
I took a glance at Google Flights and it looks like heading to Cluj will require a stop and change along the way. Flying into Bucharest gives you some choices of non-stops. Other nonstop options are Targu Mures and Brasov.
If you are over 65, public transportation in Budapest is free, including the 100E bus from the airport to Deak Ferenc ter. There is an amazing amount of information on Budapest from Mr. É on the Hungary forum. It sounds like a wonderful trip! I plan to be in Romania in September, as well, and heading back to Budapest afterwards instead of before. :)