Завтра у нас первое занятие в университете.

In Russian, the present-tense verb to be is usually omitted. So a sentence like Завтра первое занятие literally works without any word for is.

Also, Russian often expresses to have differently from English. Instead of a verb meaning we have, Russian commonly uses the pattern у + person + thing.

So:

• Завтра = tomorrow
• у нас = for us / with us
• первое занятие = first class/session
• в университете = at the university

A very literal gloss would be something like:

Tomorrow, for us, first class at the university.

But natural English is Tomorrow we have our first class at the university.

У нас literally means by us, with us, or at our place, depending on context. In many sentences, it is the normal Russian way to express possession or something that applies to a group.

Here it means something like we have.

Examples:

• У меня книга. = I have a book.
  Literally: At me, a book.
• У нас первое занятие. = We have our first class.

So in this sentence, у нас does not mean physical location like at our house. It means that this class is something scheduled for us.

Because у requires the genitive case.

The pronoun мы means we in the nominative case, but after у it changes:

• мы → нас

So:

• у меня = at me / I have
• у тебя = at you / you have
• у нас = at us / we have

This is a very common pattern, so it is worth memorizing as a unit.

Because первое agrees with занятие, and занятие is:

• singular
• neuter
• nominative

Russian adjectives must agree with the noun they describe.

Forms of первый:

• первый = masculine
• первая = feminine
• первое = neuter
• первые = plural

Since занятие is a neuter noun, the correct form is первое занятие.

This is a very common question because in English we think of have taking a direct object: we have a class.

But Russian is not using a normal transitive verb here. It is using a possession/existence structure:

у нас первое занятие

In that structure, the thing that exists or is possessed is usually in the nominative case.

So:

• У нас первое занятие.
• literally: For us, there is a first class.

That is why занятие is nominative, and the adjective первое matches it in the nominative too.

Занятие is a general word for a class, lesson, or session of study/activity.

It is similar to урок, but not always identical.

Very roughly:

• урок often means a school lesson, especially in a school context
• занятие is broader and is very common for university classes, lessons, sessions, or organized study meetings
• пара is also common in universities and refers specifically to one class period, often a double period
• лекция means a lecture
• семинар means a seminar/discussion class

So первое занятие is a very natural way to say the first class/session.

Because here в means in/at, showing location, not movement.

When в means into and shows motion toward somewhere, Russian often uses the accusative:

• Я иду в университет. = I am going to the university.

When в means in/at and shows location, Russian uses the prepositional case:

• Я в университете. = I am at the university.

So in your sentence, the class takes place at the university, which is a location:

• в университете

It is the prepositional case.

The base form is:

• университет

After в meaning in/at, it becomes:

• в университете

This is a standard use of the prepositional case for location.

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

The original sentence:

Завтра у нас первое занятие в университете.

is neutral and natural.

Other possible orders include:

• У нас завтра первое занятие в университете.
• Первое занятие у нас завтра в университете.
• В университете у нас завтра первое занятие.

These versions are not identical in emphasis:

• putting завтра first emphasizes tomorrow
• putting у нас earlier can emphasize for us / we
• putting в университете first can emphasize the location

So the meaning stays basically the same, but the focus changes.

Yes, that is possible, but it is slightly different in feel.

• Завтра у нас первое занятие в университете = Tomorrow we have our first class at the university
• Завтра наше первое занятие в университете = Tomorrow is our first class at the university

The version with у нас sounds especially natural when talking about someone’s schedule or what is happening for them.

The version with наше focuses more directly on our first class as a noun phrase.

Both are understandable, but у нас первое занятие is a very common everyday way to say we have our first class.

A helpful stress-marked version is:

За́втра у нас пе́рвое заня́тие в университе́те.

Stress:

• За́втра
• у на́с
• пе́рвое
• заня́тие
• в университе́те

A rough pronunciation guide:

• ZAF-tra
• oo NAS
• PYER-va-ye
• za-NYA-ti-ye
• v oo-nee-ver-see-TYE-tye

The most important thing is to learn the stressed syllables, because stress in Russian is not always predictable.