Ovo staro mesto još postoji.

Because they agree with mesto, which is neuter singular nominative.

• ovo = this for a neuter noun
• staro = old in the neuter singular form
• mesto = a neuter noun

In Serbian, demonstratives and adjectives must match the noun in gender, number, and case.

So:

• masculine: ovaj stari grad
• feminine: ova stara kuća
• neuter: ovo staro mesto

It is in the nominative singular.

That is because mesto is the subject of the sentence: it is the thing that still exists.

So the basic structure is:

• Ovo staro mesto = the subject
• još postoji = the predicate / verb phrase

Mesto usually means place, spot, or location.

Depending on context, it can sometimes also refer to a town or settlement, but the most basic meaning is place.

So this sentence could be understood as:

• This old place still exists.

If the intended meaning were specifically town/city, Serbian often uses grad instead.

Here još means still.

So:

• još postoji = still exists

Be careful, because još can also mean more / additional in other contexts.

For example:

• Još kafe? = More coffee?
• Još postoji. = It still exists.

The meaning depends on context.

Postoji is the 3rd person singular present tense of the verb postojati, which means to exist.

So:

• ja postojim = I exist
• ti postojiš = you exist
• on/ona/ono postoji = he/she/it exists

Since mesto is singular, the verb is singular too:

• mesto postoji = the place exists

Because Serbian has no articles like English the or a/an.

Instead, Serbian uses context, word order, and words like ovaj / ta / onaj when needed.

So English might say:

• This old place still exists
• The old place still exists

But Serbian can express definiteness without an article:

• Ovo staro mesto još postoji.

Here ovo already makes it specific: this old place.

Serbian word order is fairly flexible, but this version is the most neutral and natural:

• Ovo staro mesto još postoji.

You may see other orders for emphasis, but they can sound marked.

For example:

• Još postoji ovo staro mesto.
  This can sound like emphasizing the fact that it still exists.

The standard learner-friendly order is:

• subject + još + verb

So this sentence is a good neutral model.

Yes. That would be grammatically correct.

But the meaning changes slightly:

• Ovo staro mesto postoji. = This old place exists.
• Ovo staro mesto još postoji. = This old place still exists.

With još, the sentence suggests that its continued existence may be surprising, noteworthy, or contrasted with the past.

Because Serbian has a system of demonstratives, and ovo is the neuter form meaning this.

Very roughly:

• ovaj / ova / ovo = this
• taj / ta / to = that
• onaj / ona / ono = that over there / that one yonder

Since mesto is neuter, the correct form is:

• ovo mesto = this place

Not:

• ovaj mesto
• ova mesto

Those would be wrong.

Još is pronounced approximately like yosh.

A few helpful pronunciation points from the whole sentence:

• j = English y
• š = English sh
• c = ts
• č = ch
• ć = a softer ch
• ž = the sound in measure

So:

• ovo ≈ OH-voh
• staro ≈ STAR-oh
• mesto ≈ MES-toh
• još ≈ yosh
• postoji ≈ po-STOH-yee

Because the normal order is:

• demonstrative + adjective + noun

So:

• ovo staro mesto = this old place

This is the most natural order in ordinary Serbian.

You can think of it as similar to English:

• this old place

not usually

• old this place

Yes. Postojati is a general verb meaning to exist.

It can be used for:

• physical things: Zgrada još postoji. = The building still exists.
• places: To mesto još postoji. = That place still exists.
• abstract things: Nada još postoji. = Hope still exists.

So in your sentence, using postoji with mesto is completely natural.

Yes, in many contexts that would be a very natural English translation.

Because mesto is broad, English may translate it in different ways depending on context:

• place
• spot
• location
• sometimes even site

And još postoji can be rendered as:

• still exists
• is still there

So the exact English wording may change, even though the Serbian sentence stays the same.