Carry“Which of the two questions is better to ask the landlord: Must I pay the rent on time? or Do I have to pay the rent on time? Is there any difference, if so, what? ”
The answer is "yes" in either case, so there's no point in asking the question!
You might ask instead:
What is the grace period for paying the rent?
But to get to what your question is really about, in American English Do I have to ...? is the preferred form for asking this type of question. Occasions on which Must I ...? is needed are rare.
CJ