Translation:The driver on the bus is falling off the chair.
Is it ok to say 'falling from the chair' since in English that means falls off? Or do we absolutely have to use the 'off' because of some nuance in Hungarian?
No, there is no nuance in the Hungarian sentence. Whatever feels natural to you in English.
There are no "chairs" in buses - they are "seats". And the seat is not on the bus - it is in it. (Yes, you get on a bus, but the you sit in it. On it would be on the roof)