Learn Hardcore Croatian: Ulaznice za koncert koštaju puno. - The tickets for the concert cost a lot.

Base form: koncert (nominative singular)
After za: still koncert, but grammatically it is accusative singular (the form looks the same for many masculine nouns).

So za koncert literally means for the concert, with koncert functioning as the object of the preposition za.

Why is it Ulaznice za koncert and not something like koncertne ulaznice?

Both are possible, but they are used slightly differently:

Ulaznice za koncert = tickets for the concert (using a prepositional phrase).
Koncertne ulaznice = concert tickets (using an adjective derived from koncert).

In everyday speech, Ulaznice za koncert is very common and neutral.
Koncertne ulaznice sounds a bit more like written/advertising style (“concert tickets” as a type of ticket).

What does koštaju mean grammatically, and how is it formed?

Koštaju is the 3rd person plural, present tense of the verb koštati (to cost).
Present tense of koštati (simplified):

ja koštam
ti koštaš
on/ona/ono košta
mi koštamo
vi koštate
oni/one/ona koštaju

Since the subject ulaznice is plural, the verb is plural: ulaznice … koštaju.

Could you say Ulaznica za koncert košta puno instead? What’s the difference?

Yes:

Ulaznica za koncert košta puno. = A/one ticket for the concert costs a lot. (singular)
Ulaznice za koncert koštaju puno. = The tickets for the concert cost a lot. (plural)

The only grammatical change is:

ulaznica (singular) → košta (3rd person singular)
vs.
ulaznice (plural) → koštaju (3rd person plural).

What exactly is puno here? Is it an adjective or an adverb? Can I use mnogo instead?

In this sentence, puno functions as an adverb of quantity modifying the verb koštaju (“they cost a lot”).

You can usually replace it with mnogo with almost no change in meaning:

Ulaznice za koncert koštaju puno.
Ulaznice za koncert koštaju mnogo.

Both are correct. Puno is slightly more colloquial; mnogo can sound a bit more formal or neutral.

What’s the difference between koštaju puno and su skupe?

Both can express that the tickets are expensive, but from different angles:

Ulaznice za koncert koštaju puno. = literally “the tickets cost a lot” (focus on the amount of money).
Ulaznice za koncert su skupe. = “the tickets are expensive” (using the adjective skup, skupa, skupo = expensive).

In practice, they are often interchangeable, but koštaju puno highlights the high price, while su skupe highlights the quality of being expensive.

Why isn’t there a word for the in Ulaznice za koncert koštaju puno?

Croatian has no articles (no direct equivalents of English the or a/an).
Whether you mean the tickets or tickets is understood from context, not from a separate word.

So Ulaznice za koncert koštaju puno. can mean:

The tickets for the concert cost a lot, or
Concert tickets cost a lot,
depending on the situation.

Why don’t we need a pronoun like they before koštaju?

In Croatian, subject pronouns are usually dropped because the verb ending already shows the person and number.

English: They cost a lot.
Croatian: Ulaznice koštaju puno. (no one / oni needed)

The subject ulaznice is explicitly stated, and the verb form koštaju clearly indicates 3rd person plural, so an extra pronoun would sound redundant.

How do you pronounce koštaju and what does the j represent?

Koštaju is pronounced roughly like KOSH-tah-yu:

š = like sh in shoe
t = like t in top
a = like a in father
j = like y in yes
u = like oo in food

So -taju is pronounced -tah-yu.