U našem hotelu ima mnogo turista.

Because after u meaning in/inside, Serbian uses the locative case for location.

So:

• hotel → basic form (nominative)
• u hotelu → in the hotel (locative)
• naš → must also change to match the noun
• u našem hotelu → in our hotel

Both words change because Serbian marks grammar with endings:

• naš → našem
• hotel → hotelu

So u našem hotelu literally means in our hotel.

The preposition u can be used with different cases depending on meaning:

• u + locative = in / at a place, when talking about location
• u + accusative = into a place, when talking about movement

Here the sentence describes where the tourists are, not movement, so Serbian uses locative:

• U našem hotelu ima mnogo turista. = location

Compare:

• Idem u hotel. = I’m going into the hotel.
  (movement → accusative)
• Ja sam u hotelu. = I am in the hotel.
  (location → locative)

Yes, ima is the 3rd person singular of imati, which usually means to have.

But Serbian also uses ima in an existential sense, like English there is / there are.

So in this sentence:

• U našem hotelu ima mnogo turista.

ima does not mean has in the usual possession sense.
It means something like:

• there are
• one can find
• there exist

This is a very common Serbian structure.

Because in this existential pattern, Serbian normally uses ima in the singular, even when the thing that exists is plural.

So:

• Ima turista. = There are tourists.
• Ima ljudi. = There are people.
• Ima problema. = There are problems.

This works differently from English, where we say:

• there is one thing
• there are many things

In Serbian, ima often stays singular in this kind of sentence.

Because mnogo is a quantity word, and after quantity words Serbian usually puts the noun in the genitive.

So:

• mnogo turista = many tourists
• puno ljudi = many people
• nekoliko dana = several days

The noun turista is a masculine noun ending in -a, and its genitive plural is turista.

That’s why:

• mnogo turista is correct

not:

• mnogo turisti
• mnogo turiste

It is genitive plural.

That may look confusing, because the form turista is also the dictionary form (nominative singular). But here, because of mnogo, it is functioning as genitive plural.

A quick comparison:

• turista = a tourist (nominative singular)
• turisti = tourists (nominative plural)
• turista = of tourists / tourists after quantity expressions (genitive plural)

So in this sentence, turista is not singular in meaning. It means tourists.

Turista is one of those Serbian nouns that refer to a person and are masculine in gender even though they end in -a.

This happens with several nouns for professions or people, for example:

• turista = tourist
• kolega = colleague
• sudija = judge

So although turista looks like many feminine nouns, it behaves grammatically as a masculine animate noun.

That affects agreement and case forms.

Našem is the form of naš (our) that agrees with hotelu.

Since hotelu is:

• masculine
• singular
• locative

the adjective/pronoun must match it:

• naš hotel = our hotel (nominative)
• u našem hotelu = in our hotel (locative)

So našem is simply the matching masculine/neuter singular locative form here.

Yes. Serbian word order is more flexible than English, because case endings show the grammatical relationships.

The neutral order here is:

• U našem hotelu ima mnogo turista.

But you could also hear:

• Mnogo turista ima u našem hotelu.
• Ima mnogo turista u našem hotelu.

These versions shift the focus slightly:

• starting with U našem hotelu emphasizes the location
• starting with Mnogo turista emphasizes the quantity
• starting with Ima can sound more like introducing the situation

The original sentence is a very natural, neutral way to say it.

Yes. Puno turista is also very common and natural.

Compare:

• mnogo turista = many tourists
• puno turista = a lot of tourists

In everyday speech, puno is extremely common.
Mnogo is also very common and may sound a bit more directly like many / much.

Both are correct here:

• U našem hotelu ima mnogo turista.
• U našem hotelu ima puno turista.

Not if you want the same meaning.

• mnogo turista = many tourists / a lot of tourists
• mnogi turisti = many tourists in the sense of many specific tourists

These are not the same structure.

Mnogi turisti is a normal noun phrase with an adjective-like word (mnogi), while mnogo turista is a quantity expression.

Also, U našem hotelu su mnogi turisti would sound odd if you simply mean there are many tourists in our hotel.

For the sentence you have, ima mnogo turista is the natural choice.

Usually not in this structure.

Serbian normally says:

• Ima mnogo turista. = There are many tourists.

Using je/su would change the structure and usually would not sound natural for this existential meaning.

For example:

• Turisti su u našem hotelu. = The tourists are in our hotel.

This means you are talking about a specific group of tourists.
But:

• U našem hotelu ima mnogo turista.

means that many tourists are present/exist there, without introducing them as a definite group.

So ima is the right choice for there are here.

It is generally indefinite.

• mnogo turista means many tourists, not the many tourists
• ima in this pattern introduces existence/presence, much like English there are

So the sentence is about the presence of tourists in the hotel, not about a specific previously mentioned group.

If you wanted to talk about specific tourists, Serbian would usually use a different structure, for example:

• Turisti su u našem hotelu. = The tourists are in our hotel.

Very roughly:

• U = in
• našem = our
• hotelu = hotel
• ima = there is/exists/has
• mnogo = much/many
• turista = of tourists

So a very literal gloss would be something like:

• In our hotel there-is many of-tourists.

That sounds strange in English, of course, but it helps show how Serbian builds the sentence. Natural English is simply:

• There are many tourists in our hotel.