Negation and Quantifiers – Is ‘I Have No Much Time’ Correct?

I think the lesson on Learn English Today is very clear.

A lot of can be used in all sentences: affirmative, negative and interrogative.
Much and many are used in negative and interrogative sentences.
They are rarely used in affirmative sentences, except if they begin the sentence.

So, the sentence in the question is most commonly written or spoken as: "Paul drinks a lot of milk."

The context seems informal, and using "much" in an affirmative gives a more formal tone that sounds odd when used for a simple statement like that one.

Here is an example of using "much" in an affirmative sentence from an article on theguardian.com: "There was much talk last week of how he would be remembered for keeping Britain out of the euro[...]"

It is common to use "much" in a negative, as in: "Paul doesn't drink much milk."

or in a question, like: "How much milk does Paul drink?"

"Much" is also used with too or so, as Damkerng T. points out: "Paul drinks too much milk." or "Paul drinks so much milk!"

In general, much, as an adverb modifying adjectives, is used with a comparative adjective. Per the OED,

much, adv: modifying an expression of comparison, esp. a comparative adjective or adverb or a phrase containing a superlative adjective or adverb: in a great degree, by far

Your usage is more in line with the following colloquialism:

much, adv: modifying a positive adjective or adverb: very. Now U.S. regional (chiefly south.)

I would hesitate to call "much [adj]" ungrammatical, but it is fairly unusual with most non-comparative adjectives. When in doubt, you should probably leave off much.