Learn Spanish (Spain): Un zumo, por favor.

The word uno means one, but when it comes directly before a masculine noun like zumo, it drops the final letter to become un. You use un to mean 'a' or 'one' right before masculine nouns.

I have heard people say jugo for juice. Why is it zumo here?

In Spain, the word for juice is almost always zumo. In most of Latin America, you will hear jugo instead. Both are completely correct, but since you are focusing on Spanish from Spain, zumo is the standard word you need.

Does por favor literally mean 'for favor'?

Yes, it literally translates to 'by favor' or 'for a favor'. However, you do not need to overthink the literal translation, because it is simply the standard, everyday phrase used for 'please' in all Spanish-speaking countries.