What is the adverb here and the adjective?

Neither any nor much are adjectives. Neither is other, either.

They are Quantifiers, a type of Determiner, the same part of speech as (for instance) the, this, twenty-three, and most of the rest of. Determiners have very complex syntax in English -- an unbelievable number of the questions here are about them -- but they didn't feature much in Latin, so the Roman grammarians didn't notice them at all.

In fact, the Roman grammarians didn't even notice Adjectives, which weren't in the original Eight Parts of Speech; since Latin adjectives inflect and behave just like nouns, they were simply treated as nouns, and the extra POS slot was filled by "Participle" (a category which also accounts for a lot of ELU.SE posts).

Anyway, neither quantifiers or determiners made it into The Big Eight Parts of Speech, as touted by Big-Time Grammar Books, until quite recently. The sentence quoted in the question has no adverbs, but wholesome is certainly an adjective, modifying advice.

For a competitive exam in English grammar, what matters is not English grammar, but what the teacher or official who wrote the exam thinks counts as "English grammar". Since there are some people who will believe anything about English grammar, it follows that what we think or reply here is irrelevant.

For instance, if the question is posed by someone who believes that much "is an adverb, because it modifies an adjective", then the question is closed, for that exam. That doesn't make it a correct answer, but it makes it the right answer on that test.

Sorry.