To figure this out, it's useful to look carefully at the sentence in Spanish. (Side benefit of this approach: once you understand what's modifying what, it will be easier to translate the sentence.) Here's your original sentence again:
No te divertirás mucho
For it to be an adverb, we should look for a verb that it modifies. Candidate: "te diverterás".
for it to be an adjective, we should look for a noun that it modifies. Problem: there aren't any nouns that that we can quantify as much, many or few. The only noun here is the implied "you" ("tú"). It doesn't make any sense to modify a person with "mucho."
So, by elimination, "mucho" is modifying the verb.
A translation to English of the sentence that might help in your thinking about it might be
You won't entertain yourself a lot.
where "a lot" is an adverb that modifies "entertain yourself."
(Sure, what this means in practice is "you won't have much fun," but a translation that mirrors the original syntax more closely can make it easier to understand how the sentence is put together.)