Szeptemberben nyitva van az uszoda.

It means in. Hungarian often uses case suffixes where English uses prepositions.

• szeptember = September
• szeptemberben = in September

So this is a time expression, not a separate word like English in.

Because of vowel harmony. Hungarian suffixes often change shape depending on the vowels in the word.

A simple rule:

• words with front vowels usually take -ben
• words with back vowels usually take -ban

Szeptember has front vowels, so szeptemberben is the correct form.

Month names used as simple time expressions usually appear without an article in Hungarian.

So:

• szeptemberben = in September
• májusban = in May
• januárban = in January

That is normal Hungarian usage.

It means is open.

• nyit = to open
• nyitva = in an open state
• van = is

So nyitva van is the usual Hungarian way to say that something is open.

Because nyitva van is the standard everyday expression for something being open in a practical sense, especially for places, doors, shops, buildings, and facilities.

So Hungarian naturally says:

• Az ajtó nyitva van. = The door is open.
• A bolt nyitva van. = The shop is open.
• Az uszoda nyitva van. = The swimming pool is open.

Nyitott exists, but it is not the most natural choice here for ordinary opening/closing status.

That is true in many sentences, especially with noun or adjective predicates in the 3rd person singular:

• A ház nagy. = The house is big.
• A fiú diák. = The boy is a student.

But nyitva van is a very common state expression, and van is normally kept in a full sentence.

You may sometimes see just Nyitva on a sign, but in a normal sentence nyitva van is the standard form.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since uszoda begins with the vowel u, the correct article is az:

• az uszoda = the swimming pool

Uszoda usually means a swimming pool or swimming-pool facility—a place meant for swimming.

It is often used for a public or organized swimming place, not necessarily just any kind of pool. In some other contexts, Hungarian may use medence for pool.

Because Hungarian word order is flexible and depends a lot on topic and focus, not just subject-verb order.

Here, Szeptemberben comes first to set the time frame: in September. After that, the sentence gives the information about the pool being open.

So the sentence is structured roughly like:

• As for September, the pool is open.

This is natural Hungarian word order.

Yes. That is also correct.

The basic meaning stays the same, but the emphasis changes a little:

• Szeptemberben nyitva van az uszoda.
  → foregrounds in September

• Az uszoda szeptemberben nyitva van.
  → foregrounds the swimming pool

Hungarian often changes word order to show what the sentence is mainly about.

It is in the nominative, the basic dictionary form.

In this sentence, az uszoda is the subject:

• the swimming pool

It does not need any extra case ending here because it is not functioning as an object or location phrase.