Ο αριθμός κράτησης είναι στη λίστα επιβατών.

Word by word:

• Ο = the
• αριθμός = number
• κράτησης = of booking / of reservation
• είναι = is
• στη = in / on / at the
• λίστα = list
• επιβατών = of passengers

So the structure is basically:

• Ο αριθμός κράτησης = the booking number
• είναι = is
• στη λίστα επιβατών = on the passenger list

Because Greek often uses the genitive where English uses of.

• η κράτηση = the booking / reservation
• της κράτησης = of the booking
• κράτησης on its own can mean of booking / reservation in this kind of noun+noun combination

So:

• αριθμός κράτησης literally = number of booking
• natural English = booking number or reservation number

This is a very common Greek pattern:

• στάση λεωφορείου = bus stop, literally stop of bus
• πόρτα αυτοκινήτου = car door, literally door of car

For the same reason: it is also a genitive form.

• ο επιβάτης = the passenger
• οι επιβάτες = the passengers
• των επιβατών = of the passengers

In the sentence:

• λίστα επιβατών = list of passengers
• natural English = passenger list

So Greek is using two noun phrases built with the genitive:

• αριθμός κράτησης = booking number
• λίστα επιβατών = passenger list

Greek uses the definite article much more often than English.

Here, Ο αριθμός means the number. In this sentence, it refers to a specific booking number, so the article is natural and expected.

If you removed it and said just Αριθμός κράτησης, that would usually sound less like a full normal sentence and more like:

• a heading
• a label on a form
• a dictionary-style phrase

So in a complete sentence, Ο αριθμός κράτησης is the normal choice.

στη is the contracted form of:

• σε
  • τη(ν) = στη(ν)

So:

• σε τη λίστα becomes στη λίστα

This is extremely common in modern Greek.

Examples:

• στο σπίτι = σε το σπίτι = in the house / at home
• στην πόλη = σε την πόλη = in the city
• στους φίλους = σε τους φίλους = to the friends

In your sentence:

• στη λίστα = in/on the list

Because the preposition σε takes the accusative case in modern Greek.

The noun λίστα is feminine, and its accusative singular form is the same as its nominative singular form:

• η λίστα = nominative
• τη λίστα = accusative

So:

• στη λίστα = σε τη λίστα

Even though the article changes form, the noun itself looks the same here.

This is a good example of languages not matching word-for-word.

Greek σε / στη can cover meanings that English expresses with in, on, at, or to, depending on context.

So:

• στη λίστα literally might look like in the list
• but the natural English translation is on the list

This is normal. You should learn στη λίστα as the standard Greek expression, not try to force a one-to-one preposition match.

In modern Greek, είναι is normally used in a sentence like this.

So:

• Ο αριθμός κράτησης είναι στη λίστα επιβατών. = normal full sentence

If you remove είναι, you get:

• Ο αριθμός κράτησης στη λίστα επιβατών

That does not sound like a normal complete sentence. It sounds more like a fragment, a note, or an incomplete phrase.

So yes, here είναι is needed.

αριθμός is in the nominative singular because it is the subject of the sentence.

Breakdown:

• Ο αριθμός κράτησης = the subject
• είναι = the verb
• στη λίστα επιβατών = where it is

So the core structure is:

• The booking number
  • is
    • on the passenger list

That is why αριθμός is nominative, while κράτησης and επιβατών are genitive.

Yes. Greek word order is more flexible than English, although the original sentence is very natural.

Standard neutral order:

• Ο αριθμός κράτησης είναι στη λίστα επιβατών.

You could also say:

• Στη λίστα επιβατών είναι ο αριθμός κράτησης.

But that sounds more marked or emphatic, as if you are emphasizing where the booking number is.

For most learners, the original order is the best one to use.

A simple pronunciation guide:

• Ο αριθμός κράτησης είναι στη λίστα επιβατών
• roughly: o a-ri-THMOS kra-TI-sis EE-ne sti LEE-sta e-pi-VA-ton

A few helpful points:

• θ sounds like th in think
• αι in είναι is pronounced e
• η, ι, υ, ει, οι are all pronounced like ee in modern Greek
• stress matters a lot in Greek:
  • αριθμός
  • κράτησης
  • είναι
  • λίστα
  • επιβατών

It is a very normal and correct way to say it, but not the only possible expression.

Depending on context, you may also see:

• κωδικός κράτησης = booking code
• αριθμός κράτησης = booking number
• sometimes in travel contexts, terms influenced by English may also appear

So the phrase in your sentence is perfectly natural, especially in formal or travel-related Greek.

Because in Greek, when one noun modifies another in a fixed phrase, the second noun often appears in the genitive without its article.

So:

• ο αριθμός της κράτησης = the number of the booking
  This sounds more specific: a particular booking

But:

• ο αριθμός κράτησης = booking number / reservation number
  This is more like a standard label or type of number

Similarly:

• η λίστα των επιβατών = the list of the passengers
• η λίστα επιβατών = passenger list

Both can be correct, but the version without the article is often more compact and idiomatic in this kind of expression.