Szeptemberben hideg a víz.

Szeptemberben means in September.

The ending -ban/-ben is a Hungarian case ending that often corresponds to English in. Instead of using a separate preposition, Hungarian usually adds a suffix to the noun.

So:

• szeptember = September
• szeptemberben = in September

A few similar examples:

• januárban = in January
• iskolában = in the school
• városban = in the city

With months, -ban/-ben is the normal way to say in a given month.

This is because of vowel harmony, a very important feature of Hungarian.

Hungarian suffixes often have two versions, and the form depends on the vowels in the word.
Here the pair is:

• -ban
• -ben

Since szeptember contains front vowels, it takes -ben:

• szeptemberben

Compare:

• januárban
• májusban
• decemberben
• októberben

So the choice is not random; it follows the sound pattern of the word.

In Hungarian, the verb to be is usually left out in the present tense, third person singular when you are simply saying that something is something.

So:

• A víz hideg. = The water is cold.
• literally: The water cold.

That is normal Hungarian grammar.

You do use the verb in other situations, for example:

• A víz hideg volt. = The water was cold.
• A víz hideg lesz. = The water will be cold.
• Van víz. = There is water.

So in your sentence, the missing is is not a mistake; it is exactly how Hungarian works.

Hungarian often uses the definite article a/az in places where English would use no article.

In Szeptemberben hideg a víz, a víz means the water. This can refer to water that is understood from context, or to the water as the relevant thing being discussed.

English often says:

• Water is cold in September
• or The water is cold in September

Hungarian naturally prefers:

• Szeptemberben hideg a víz.

So you should not expect article usage to match English exactly.

Hungarian has two forms of the definite article:

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

Since víz begins with v, which is a consonant, the correct form is:

• a víz

Compare:

• a víz = the water
• a ház = the house
• az alma = the apple
• az idő = the weather / the time

So this is similar to the difference between a and an in English, although the rule is not exactly the same in every detail.

Yes, A víz hideg is also correct.

Hungarian word order is more flexible than English word order. It often reflects topic and emphasis, not just grammar.

In Szeptemberben hideg a víz, the word Szeptemberben sets the time frame first, and then hideg a víz gives the main statement. This sounds natural and idiomatic.

You can also say:

• A víz hideg. = The water is cold.
• A víz szeptemberben hideg. = The water is cold in September.
• Szeptemberben a víz hideg. = also possible

These versions are all grammatical, but they do not sound exactly identical in emphasis. Hungarian word order is often about what you are talking about first and what you are presenting as the main information.

Yes: in Hungarian, month names are normally lowercase.

So ordinarily you would write:

• szeptember
• szeptemberben

In your sentence, it is capitalized only because it is the first word of the sentence:

• Szeptemberben hideg a víz.

If the word appeared in the middle of a sentence, it would usually be lowercase.

Yes, hideg is an adjective, and here it is used as a predicate adjective:

• A víz hideg. = The water is cold.

You do not use hidegen here, because hidegen is generally an adverbial form, meaning something more like coldly or in a cold state/way, depending on context.

Compare:

• A víz hideg. = The water is cold.
• Hidegen beszélt. = He/she spoke coldly.

So in your sentence, hideg is the correct form.

A rough pronunciation guide is:

• Szeptemberben ≈ SEP-tem-ber-ben
• hideg ≈ HEE-deg
• a ≈ a short aw/o-like sound
• víz ≈ veez

A few useful pronunciation points:

• sz is pronounced like English s
• Hungarian s by itself is pronounced like English sh
  
  • so sz and s are not the same
• í is a long vowel, so víz has a long ee sound
• Stress in Hungarian normally falls on the first syllable of each word

So the sentence is stressed roughly like this:

SZEP-tem-ber-ben HI-deg a VÍZ