Learn Spanish (Latin America): El invitado bebe muy poco.

In this sentence, poco is an adverb describing the verb bebe (how much he drinks). Adverbs never change their ending to match gender or number. If it were describing a noun, like saying "few guests" (pocos invitados), it would change, but here it always stays as poco.

Could I say mucho poco to mean "very little"?

No, you have to use muy (very). Mucho means "a lot" or "much." When you want to say "very" before an adjective or another adverb (like poco), you use muy.

Does this mean he drinks "a little bit" or "hardly anything"?

It means he drinks "little" (hardly anything). In Spanish, just like in English, there is a big difference between poco (little / not much) and un poco (a little bit). Saying bebe muy poco emphasizes that the amount is noticeably small.

If the guest was a woman, would the sentence change?

Yes, the word for "guest" changes depending on gender. For a female guest, you would say La invitada bebe muy poco. Notice that muy poco stays exactly the same, because adverbs don't change to match the gender of the person!