Learn Finnish: Kukaan ei saa yhtään voittopistettä.

Because this is a negative sentence, the personal ending is attached to the negative verb (here, ei marks the 3rd person singular). The main verb is stripped down to its present connegative form, which is just the raw stem. For the verb saada, that stem is saa, leaving it with zero suffixes. If this were a complex positive sentence, you would see the slots fill up according to the rules, for example: saataisiinkohan (stem saa + passive -ta- + conditional -isii- + person -n- + clitics -ko-han).

Does kukaan ei saa literally mean "nobody doesn't get"?

Essentially, yes! Finnish uses negative concord, meaning you must pair negative pronouns with a negative verb. The word kukaan means "anyone" in a question, but paired with ei it becomes "nobody". You cannot say kukaan saa to mean "nobody gets".

Is yhtään related to the number one (yksi)? How does it mean "any"?

Historically, yes. It is built from the partitive of yksi (yhtä) plus the enclitic particle -än. In modern Finnish, however, yhtään functions as a set vocabulary word meaning "any" (before a noun) or "at all" (when modifying a verb). It is almost exclusively used in negative sentences and questions.

Is voittopistettä in the partitive case because it comes after yhtään?