Learn English – Is it grammatically correct to say ‘I have not much time?’

I think as most of the sayings go back to the olden days, old English, other languages, or different dilects, some of them seem to be incorrect grammatically.

The subject saying isn't common in use, it was used especially by some famous politicians, John F. Kennedy being one of them.

I think the OP is right that we should use don't ask instead of "ask not" to make the saying modern in English, but if you change the words of the saying, it'll spoil its beauty.

The phrase no one of them is "grammatical" (taking that to mean that it is both idiomatic and meaningful), but it is not always the same thing as none of them. It is an emphatic form employed to distinguish a proposition concerning a number of entities taken singly from propositions concerning the same entities taken in combination.

Consider the difference between these two propositions concerning three people, Adolphus, Benjamin and Carloman, suspected of murdering Dionysius.

1. None of them killed him. This is taken to assert that all three are innocent of the murder.

2. No one of them killed him. This implies the possibility that two of them—A and B, or A and C, or B and C—or all three, conspired to commit the murder.

So it might be perfectly proper to say of three utterances that "No one of them is grammatical" if you intend to propose that some combination of the utterances is grammatical.