Ne znam cenu ove jakne.

Because cenu is the accusative singular form of cena.

The noun cena means price, and in this sentence it is the direct object of znam (I know / I don’t know), so Serbian uses the accusative:

• cena = nominative, dictionary form
• cenu = accusative singular

So:

• Znam cenu. = I know the price.
• Ne znam cenu. = I don’t know the price.

English does not change the noun here, but Serbian does.

Because ove jakne means of this jacket, and after cena Serbian normally uses the genitive:

• cena jakne = the price of the jacket
• cena ove jakne = the price of this jacket

Here:

• jakna is the basic form
• jakne is the genitive singular
• ove matches jakne in gender, number, and case

So the structure is:

• cenu = the price
• ove jakne = of this jacket

Because ovu would be the accusative singular feminine, but here the phrase is not functioning as the direct object.

Compare:

• Vidim ovu jaknu. = I see this jacket.
  Here ovu jaknu is the direct object, so accusative is used.

But in your sentence:

• Ne znam cenu ove jakne.

The direct object is cenu, not jaknu.
The phrase ove jakne depends on cenu and means of this jacket, so it goes into the genitive.

That is why you get:

• ova = nominative
• ovu = accusative
• ove = genitive/dative/locative singular feminine

Not here. The object still stays in the accusative:

• Znam cenu.
• Ne znam cenu.

A learner coming from Russian may expect negation to change the object to genitive, but in modern Serbian that is not what happens in a sentence like this.

So ne znam does not explain ove jakne.
The genitive there comes from cena + genitive, not from negation.

The dictionary forms are:

• ne = not
• znam → from znati = to know
• cenu → from cena = price
• ove → from ovaj / ova / ovo = this
• jakne → from jakna = jacket

So if you looked these up in a dictionary, you would usually search for:

• znati
• cena
• ovaj
• jakna

No, poznajem would not be right here.

Serbian usually distinguishes between:

• znati = to know a fact, piece of information, how to do something
• poznavati / poznajem = to know a person, place, or be familiar with

So:

• Znam cenu. = I know the price.
• Poznajem Marka. = I know Mark.

Since price is information, Serbian uses znati, not poznavati.

Because Serbian does not have articles like English the and a/an.

English says:

• I don’t know the price of this jacket.

Serbian simply says:

• Ne znam cenu ove jakne.

Definiteness is understood from context, and in this sentence ove (this) already helps make it specific.

So Serbian often expresses the same idea without an article.

Yes, Serbian word order is more flexible than English, but the original sentence is the most neutral and natural:

• Ne znam cenu ove jakne.

You can move things around for emphasis, for example:

• Cenu ove jakne ne znam.

This gives extra focus to cenu ove jakne.

Still, for a learner, the best default word order is the original one.

You know it is feminine because its basic form ends in -a, which is very often a sign of a feminine noun in Serbian.

That matters because words that go with it must agree with it:

• ova jakna = this jacket
• ove jakne = of this jacket
• ovu jaknu = this jacket (as object)

The demonstrative changes form depending on the noun’s:

• gender
• number
• case

So the feminine gender of jakna is the reason you see forms like ova, ove, and ovu.

Here it means this.

The form ove can be confusing because it can also mean these in other contexts.
You have to look at the noun form that follows it.

In this sentence:

• ove jakne

means of this jacket, because jakne is genitive singular, not nominative plural.

Compare:

• ova jakna = this jacket
• ove jakne = of this jacket
• ove jakne = these jackets

So the form alone is not always enough; the grammar of the whole phrase tells you the meaning.

A rough English-style pronunciation is:

neh ZNAHM TSEH-noo OH-veh YAHK-neh

A few important sounds:

• c = ts as in cats
• j = y as in yes
• nj = a soft ny sound, similar to canyon
• e is usually pronounced clearly, not reduced to a weak sound like English uh

So:

• Ne ≈ neh
• znam ≈ znahm
• cenu ≈ tseh-noo
• ove ≈ oh-veh
• jakne ≈ yahk-neh

Yes. The given sentence is completely natural, but Serbian might also use other wording depending on context. For example:

• Ne znam koliko košta ova jakna. = I don’t know how much this jacket costs.

This is slightly more like everyday spoken English I don’t know how much this jacket costs.

Your original sentence, though, is perfectly good and very common, especially when focusing on the price as a noun:

• Ne znam cenu ove jakne.