Learn Hardcore Finnish: Oppitunti kestää vain vartin tänään. - The lesson lasts only a quarter of an hour today.

Here kestää means “to last” or “to take (time).” When you express how long something lasts in Finnish, you use kestää followed by a time expression in the partitive case.
Example: Oppitunti kestää vartin = The lesson lasts a quarter of an hour.

Why is vartin in the partitive case instead of the nominative vartti?

Finnish uses the partitive case for durations and amounts to show an indefinite or ongoing quantity. Since “a quarter of an hour” is a span of time (not a fixed, countable object), vartti becomes vartin.

What function does vain serve here? Can it be placed in a different position?

Vain means “only” or “just,” adding limitation to the duration. It usually comes immediately before the word it modifies (vain vartin). You can shift it for emphasis:

Vain oppitunti kestää vartin tänään. (Only the lesson…)
Oppitunti kestää vartin vain tänään. (Lasts a quarter of an hour only today.)

Why is tänään at the end, and could it go elsewhere?

Finnish allows flexible word order. Putting tänään (“today”) at the end emphasizes when it happens. You can also say:

Tänään oppitunti kestää vain vartin. (Focus on “today”)
Oppitunti tänään kestää vain vartin. (Slightly more marked, but still correct)

Could I replace vartin with 15 minuuttia? How would that change the sentence?

Yes. Just remember minuutti also takes partitive in duration contexts:
Oppitunti kestää vain 15 minuuttia tänään.
Meaning remains “The lesson lasts only 15 minutes today.”

Is oppitunti singular here? How do I say “the lessons” in plural?

Yes, oppitunti is singular (“the lesson”). For multiple lessons, use plural nominative oppitunnit, and match the verb:
Oppitunnit kestävät vain vartin tänään.
(“The lessons last only a quarter [each] today.”)