Learn Hardcore Finnish: Se kestää vain hetken.

Se kestää tunnin. (It lasts an hour.)
Se kestää viikon. (It lasts a week.) So hetki (moment) becomes hetken (“for a moment”). The nominative hetki is not used for this duration meaning.

Which case is "hetken" exactly—genitive or accusative?

Morphologically it’s the genitive singular of hetki. In this time-measure use, grammars often call it an “adverbial genitive.” You may also see it analyzed as the -n “accusative,” but for learners it’s best to remember: genitive form for a specific, bounded duration.

Could I say "Se kestää vain hetkeä" or "Se kestää vain hetki"?

Se kestää vain hetki is ungrammatical; nominative doesn’t work for duration.
Se kestää vain hetkeä (partitive) isn’t idiomatic here. Partitive suits unbounded amounts, e.g. Se kestää tunteja (“It lasts for hours”). For “a (single) moment,” use hetken.

What does each word correspond to?

Se = it / that (3rd-person singular pronoun)
kestää = lasts, takes (3rd-person singular present of the verb kestää “to last; to take (time); to endure”)
vain = only, just
hetken = a moment (genitive singular of hetki)

How is kestää conjugated (present and a couple of key forms)?

Present: minä kestän, sinä kestät, hän/se kestää, me kestämme, te kestätte, he kestävät
Past: kesti
Perfect: on kestänyt
Negative (3sg): ei kestä

Does kestää always mean “to take (time)”?

No. It also means “to last” (duration/durability) and “to endure/tolerate” (transitive):

Esitys kestää tunnin. (The performance lasts an hour.)
Silta ei kestä painoa. (The bridge can’t bear the weight.)
En kestä melua. (I can’t stand noise.)

Can I drop se and just say Kestää vain hetken?

Yes. Finnish often omits an impersonal “it.” Kestää vain hetken is natural when context makes the subject obvious.

Why use se instead of tämä or tuo?

Se refers to something already known in the context and is the default neutral choice. Tämä = “this (near me/just mentioned)” and tuo = “that (near you/over there).” You can say Tämä kestää vain hetken to emphasize “this one” specifically.

Where should vain go? Can I say Se vain kestää hetken or Vain se kestää hetken?

Place vain immediately before what it limits.

Se kestää vain hetken. = Only the duration is small. (neutral)
Se vain kestää hetken. = “It just/merely lasts a moment” (slightly different pragmatic feel).
Vain se kestää hetken. = Only that one lasts a moment (others don’t).

Is vain the same as vaan?

In standard Finnish, vain means “only.” Vaan means “but (rather)” after a negative: En tätä, vaan tuon. In casual speech/writing, people often use vaan where standard Finnish would have vain: Se kestää vaan hetken. That’s colloquial.

How do you pronounce the sentence?

Se: [se] (short e)
kestää: [ˈkes.tæː] (long ää, like the a in “cat,” but longer)
vain: [ʋɑin] (diphthong “ai” like English “eye”; Finnish v is between v and w)
hetken: [ˈhet.ken] (clear t and k) Primary stress is on each word’s first syllable: SE KES-tää VAIN HET-ken.

Are there other natural ways to say “It only takes a moment” in Finnish?

Yes:

Siihen menee vain hetki.
Se vie vain hetken.
Se on ohi hetkessä.
Se kestää vain pienen hetken.

Why is there no word for “a” in “a moment”?

Finnish has no articles. Indefiniteness/definiteness is conveyed by case, word order, quantifiers, and context. hetken here already implies “a single moment.”

Is hetkinen related? And what about Hetkinen! as an interjection?

Yes. hetkinen is a diminutive-like form meaning “a short moment,” and Hetkinen! is a common interjection: “Just a moment!” As a time object you might say odota hetkinen (“wait a moment”), but with kestää the usual form is hetken.

I’ve heard Ei kestä! meaning “You’re welcome.” Is that the same verb?

Yes. Ei kestä! is a fixed polite reply to thanks, roughly “Don’t mention it.” Literally it’s “(it) doesn’t last/it’s not worth (thanking),” using the negative ei + kestä (3rd-person).