You are bound to get a wide range of answers for each question about English. There are so many possibilities, you are likely to learn a lot, you are bound to learn a lot. I wouldn't use "bound" in the sense of "have to" in the hotel sentence. I am bound by honour to tell the truth, I suppose.
Oh.... Thanks for all your comments... Now I am totally confused... LOOOOL So when should I use "be bound to" and when should I use "be supposed to". Are they interchangeable? Or I just need to learn for which sentence which is better?
If you mean "should", use "supposed to". ("Bound" would sound too formal, even archaic.)
If you want to use "bound", save it for situations like this: - "The job is bound to take longer than you think."---LONGMAN English-Japanese Dictionary.- Something is going to happen or is already set in motion and you will probably not be changing it.
To summarise a few other answers (and add my own two cents), they are not direct synonyms and 'supposed to' is the more useful/common expression.
'Bound to' can have a couple of different meanings (maybe more).
In all of Colin's three examples in the post above, 'bound to' means 'likely to'.
But in your original sentence, bound is the past participle of bind. 'He is bound to do his homework' indicates to me some kind of pact or agreement that he has already made that will force him to do the homework. Often you are bound to something/someone because you have promised to do something or because you share something special with someone. In the Lord of the Rings films, Frodo says something to Sam at one point, talking about Gollum, like, "He is bound to me, and I to him." The ring binds Frodo and Gollum together.