I was there in 1997, lots of Westernization was coming: chicken McNuggets, the Ford Fiesta and Pantene shampoo were all “Uj!” (new!) that year.
I didn’t say in a hotel (had friends who let us stay in their place while they were in Russia) so I couldn’t tell you about that.
On the Buda side of the river there is a cool art/museum/castle at the top of the hill. You can take a funicular lift (a sort of cable car) up the hill which is very cool and gives good views of the downtowny/Parliament area.
If you like geeky stuff, seek out the Museum of Electrotechnology. It was one of the coolest things I did in 7 weeks of Europe travel. Basically a bunch of retired physics professors, who get like 1 visitor a day, explaining how Hungary was at the center of electric technology. Neat exhibits, even if their English is a little rusty. (or was). It’s located on a dank side street not too far from the Chain Bridge (IIRC) on the Pest side.
You can take a cheap commuter rail to Szentendre (a vacation town) and then a ferry to even smaller towns on the Danube, if that’s your thing. Szentendre has a Museum of Marzipan that’s worth seeing just for the life-sized Michael Jackson made of marzipan candy. In Estergom (one of those tiny towns up the Danube), the big church has a museum of papal vestments, and the entire thighbone of St. Steven in a reliquary. (as you might imagine I go in for strange little museums)
I didn’t find the subway at all terrifying (but then I grew up in New York City) and the people are nice, if a bit dour. The main language of tourism is actually German, not English, so if you can pick up a few words it smoths the way. I speak about 10 words of German: 1, 2, 3, good, bad, night, day, please, thank you, yes, no. They came in handy.
Hungarian is so astonishingly difficult to learn that if you learn to say just please and thank you during your visit, Hungarians will be impressed.
Hungary has some a local fruit brandy called Palinka (comes in plum, pear, peach, etc.) and a truly vile herbal liquer called Unicum. If you want your bottled water without bubbles its “sin gaz.”