The usual expressions you hear are:
You mustn't stay up late.
You have to not state up late.
These mean that it is forbidden to stay up late.
You don't have to stay up late.
You don't need to stay up late.
You needn't stay up late.
These mean that it is not necessary to stay up late.
This leaves your statement 3:
I haven't to stay up late.
As the comments say, "I have not to" is an incredibly rare expression in English. If I've heard anybody use "I have not to", I have forgotten it. But given the rules of English grammar, it should mean the same thing as "I don't have to".
The only difference between "I have not to" and "I don't have to" is that one uses do-support and the other doesn't. In English, adding do support can give a statement emphasis, but it doesn't change the meaning. In the same way that "I haven't a clue" means the same thing as "I don't have a clue", wouldn't the natural assumption be that "I haven't to go" means "I don't have to go"?
Of course, if "I haven't to" was an idiom, the idiomatic usage could overrule this natural assumption. But nobody uses it, so it's not an idiom. So I have to believe that if it means anything, it means "I don't have to".
UPDATE: In the book Survey of English Dialects, found through Google books, it says that "I haven't to" means "I mustn't" in the Du dialect of English. So in some region of England, it is an idiom. Outside of that region, I would assume that it means I don't have to, which is what it seems to mean in the majority of the instances I have found through Google books. I can't be sure what Du means, since the key isn't on one of the previewable pages but it's probably Durham, a city and a county in northern England.