Learn English – Is it right to say “on the drive”

Both prepositions are correct but have slightly different meanings here, depending on how the author considers the bus. The interpretation also depends on context¹.

"On the bus" considers the bus functionally as a form of transport.

"In the bus" emphasises that the bus is a place.

So if I read that someone "fell asleep in the bus", my first impression is that the bus is not in use (maybe it is abandoned somewhere, or maybe the character in the novel broke into the bus company's parking lot and got on a bus at night).

If I read instead that someone "fell asleep on that bus", I imagine it to be a bus that is in use as transportation, so the character caught the bus and fell asleep while it travelled to its destination.

(so far as I can think, this use of on is limited to forms of transportation. One can be on a bus, on a ship or on a plane, while actually being inside. As others have pointed out, if you said you were "on that house" you would be standing on the roof.

1. It is possible to be on a bus that is not in service and vice versa, but that is unusual and requires additional context.

on the house (Cambridge dictionaries online)
If you have something on the house, it is given to you free by a business

It's on rather than in because the idiomatic usage is nothing to do with physical location (for which contexts in is more common in general). It's more a matter of on whose shoulders does the burden of cost fall?, or on whose bill does that cost appear?. The usage also commonly appears in forms like...

"Let's have another drink! This one's on me!". (i.e. - I will pay for it)

OP's on the house effectively combines two idiomatic usages - where on X means cost to be borne by X, and the house is the commercial establishment currently providing your food / drink / entertainment.

Note this closely-related idiomatic usage where the choice of preposition is rather less fixed...

They live on/off State benefits (US: on/off welfare)

Both AmE and BrE favour on in the above, but off is also quite common. Some native speakers may always use one or the other preposition in such constructions, and some may see them as synonymously interchangeable. But to me at least, on there carries more "neutral" implications of a low level of income, whereas off is a more "loaded" usage (implying criticism of people who can't/won't fend for themselves).