The cute little cable car has been transporting millions of passengers from the Chain Bridge to the top of the Castle hill (or the other way) since its opening in 1870. You might have seen it appearing in Wes Anderson’s The Grand Budapest Hotel, too.
Originally it was operated by steam and was destroyed in WW2. Since its reconstruction it’s been an electric cable car.
My insider tip: if you want to avoid long lines, don’t take it to go to the top but take it to descend to the Chain Bridge from the Royal Palace.
Buda Castle Funicular 
Buda Castle Funicular