You are staying 2.5 days in a 4-day city so smooth movement throughout the city is going to be a bit more important. I say smooth because no one wants to hurry or go on forced tourist marches. Just smooth and somewhat seamless.
So how many rides indeed. Enjoyment of walking, interests, and especially where you are staying those three nights will dictate that. If you are staying in Buda, then twice as many rides. If you are staying in District V is possibly the fewest number but at another cost. If it’s Viking cruise, you might be at the Four Seasons then it’s more tickets again. A pretty day, fewer tickets, a morning shower, more tickets. I live here and I can’t guess.
Then a pass vs a ticket. Let’s at least assume that you will buy a block of 10 single tickets at a time, in which case it will cost you 450 Ft a ride. That’s about $1.35 a ride.
You mentioned the BudapestGo App and if that is your preference then: at each tram the four of you will wait for the tram to arrive, stand outside the door and using your phone scan the QR code. There won’t be too many others trying to do it because most in the city use passes, not tickets. You will have 10 to 15 seconds between the time the tram stops until the door closes again for the four of you to get this done and board the tram. If you miss getting on you can’t use that scan for the next tram. If you use paper tickets, you just get on and then find a validation machine. Most of them work, but not all, so you may have to find more than one. If the tram is crowded do push through because the fine is high if you get caught without a validated ticket a broken machine or crowded tram is no excuse.
The 72-hour card is a tourist card, and the city has no qualms with making the tourist subsidize the locals. It’s a poor country after all. So, the 5750 Ft card is the equivalent of 12.78 tickets.
Will you ride the transportation 12.78 times in 2.5 days? If you have taken the time to learn the system a bit, if you want to feel local, if you want smooth days ... maybe so. It wouldn’t surprise me. It’s only 5.11 rides a day. From the Four Seasons Hotel to the Market Hall to the Opera to Nyugati to the parliament and back to the hotel is 4 tram rides; but then after you are going out for dinner and that might be a few more tickets.
So, how does the card work? You carry it. Period, end of explanation. If an inspector should ask you, you show it. No QR codes, no validation, just hop on and hop off at your hearts content.
Let’s say you do the math and determine that you will only use 10 tickets in 2.5 days. Those 10 tickets will cost you 4.500 Ft. That will save you 1250 Ft ($3.75 or about a buck-fifty a day) compared to a 72 hour card; but for the convenience and insurance if your count is wrong, I would buy the 72-hour pass.
If you want to save real money, ride the 100E bus to and from the airport. Or for a bigger savings the 200E.