Learn Hardcore Finnish: Tämä käynti virastossa kesti vain vartin. - This visit to the office lasted only a quarter of an hour.

virastossa = in/at the office (where you were)
virastoon (illative) = into/to the office (movement/destination)
So käynti virastossa is “a visit at the office.” If you wanted to stress going there, you’d more naturally say something like Kävin virastossa (“I went to the office / visited the office”), but with the noun käynti, the place is commonly given as the location of the visit.

What case is käynti in, and why?

Käynti is in the nominative singular. Here it’s the subject of the sentence: the thing that lasted.
The full subject phrase is Tämä käynti virastossa (“This visit at the office”).

What does tämä do here, and does it affect the noun form?

Tämä means “this” and agrees with the noun in number and case. Since käynti is nominative singular, tämä is also nominative singular: tämä käynti.

Why is the verb kesti used, and what form is it?

Kesti is the simple past (imperfect) of kestää (“to last”). It’s used because the sentence describes a completed past event: the visit has already happened.

Why is it vain vartin and not vain vartti?

Because Finnish commonly expresses duration with the partitive (especially in answers to “how long?”).

vartti = nominative “a quarter (of an hour)”
vartin = genitive of vartti, and in this specific expression it functions as a measure phrase meaning “(for) a quarter of an hour.”

In everyday Finnish, vartin is the normal way to say “for fifteen minutes / a quarter of an hour” with kestää:

Se kesti vartin. = “It lasted (for) a quarter of an hour.”

So is vartin genitive or accusative here?

Formally it looks like genitive singular (vartin), and it’s usually taught/understood as a genitive measure expression (“a quarter’s worth [of time]”).
You may also see explanations that treat some time-measure objects with past-tense verbs as “total object”-like, but for a learner it’s most practical to remember this as a fixed duration expression: kesti + vartin/tunnin/viikon etc.

Is vain always placed before the thing it modifies?

Very often, yes. Vain (“only/just”) usually goes directly before the word/phrase it limits:

kesti vain vartin = “lasted only a quarter-hour”
You could also say Tämä käynti virastossa kesti vartin vain, but that sounds more marked/emphatic.

Why is the word order Tämä käynti virastossa kesti vain vartin—can it change?

Yes, Finnish word order is flexible, but the neutral order here is:
Subject + verb + adverb + duration.

You could also front things for emphasis:

Vain vartin tämä käynti virastossa kesti. (emphasis: “only a quarter-hour”)
Virastossa tämä käynti kesti vain vartin. (emphasis on “at the office”)
The meaning stays basically the same, but the focus changes.

Why is there no word for “for” (as in “for fifteen minutes”)?

Finnish often expresses “for” in time durations without a preposition, using case forms or set constructions. With kestää, you typically just add the duration:

kesti vartin / tunnin / kolme tuntia = “lasted (for) 15 min / an hour / three hours”

Could I replace vartin with 15 minuuttia? What case would it be?

Yes. The most common is:

kesti vain 15 minuuttia
Here minuuttia is partitive singular (a partitive “measure” form). With numbers other than 1, Finnish uses partitive singular: 15 minuuttia, 3 tuntia, etc.

What exactly is virastossa: “in the office” or “at the office”?

Inessive (-ssa/-ssä) covers both “in” and “at” in many everyday location contexts. virastossa can be translated as either depending on what sounds natural in English. It basically means “at the office (premises), inside the office building.”

Does this sentence imply a single visit, or could it be general (“this kind of visit”)?

It strongly implies one specific visit, because of tämä (“this”) and the past tense kesti. For a general statement you’d more likely use something like:

Käynti virastossa kestää yleensä vain vartin. (“A visit to the office usually lasts only 15 minutes.”)

Is käynti the same as vierailu? Why choose one over the other?

They’re close, but not identical:

käynti = a practical “going/stop/errand/visit,” often routine (doctor’s, office, shop).
vierailu = “visit” more like visiting people/places as a guest, often more social or formal.
For dealing with an office, käynti virastossa sounds very natural.