Az időkorlát egy óra, és a visszaszámlálás elkezdődik.

Hungarian has two forms of the definite article:

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

So:

• az időkorlát because időkorlát begins with i
• a visszaszámlálás because visszaszámlálás begins with v

This is a bit like English a/an, except in Hungarian this alternation happens with the definite article, not the indefinite one.

Because it is a Hungarian compound noun. Hungarian very often joins words together where English would use two separate words.

• idő = time
• korlát = limit

So időkorlát is literally time-limit. Writing it as one word is normal Hungarian spelling.

In Hungarian, the verb van (is/are) is usually omitted in the present tense, third person, when the predicate is a noun or adjective.

So:

• Az időkorlát egy óra
  literally looks like The time limit one hour
• but it means The time limit is one hour

This is one of the most important basic patterns in Hungarian.

It can mean either, because Hungarian uses the same word, egy, for both:

• the number one
• the indefinite article a/an

Here, egy óra is best understood as one hour, though in many contexts it can also simply correspond to English an hour.

Because óra here is not an object. It is the predicate noun in a sentence with an omitted is.

So Hungarian keeps it in its basic form:

• Az időkorlát egy óra = The time limit is one hour

You would use órát only when óra is a direct object:

• Várok egy órát = I wait for an hour

So the lack of an ending here is completely normal.

It is another compound-like formation:

• vissza = back
• számlálás = counting

So visszaszámlálás literally means something like counting back, which is why it means countdown.

Hungarian often builds words this way, so breaking them into parts can really help.

El- is a verbal prefix. In many verbs it helps express that an action gets underway or reaches a clear starting point.

So:

• kezdődik = begins / is beginning
• elkezdődik = starts / begins, with a stronger sense that the action actually sets in

In this sentence, elkezdődik sounds very natural for the countdown begins.

Because the two verbs behave differently:

• elkezd is usually transitive: someone starts something
• elkezdődik is intransitive: something starts / begins

Here the subject is a visszaszámlálás (the countdown), and it is the thing that begins, so elkezdődik is the right form.

Compare:

• Elkezdem a munkát. = I start the work.
• A munka elkezdődik. = The work begins.

Yes, both are possible.

Hungarian word order is much more flexible than English word order. It often depends on what is treated as the topic or what is being emphasized.

• A visszaszámlálás elkezdődik is a neutral, topic-first way to say it.
• Elkezdődik a visszaszámlálás is also natural and sounds a bit more like announcing the event itself.

So the version in your sentence is not the only possible one, but it is perfectly normal.

Yes. It is grammatically present tense.

Hungarian often uses the present tense for:

• scheduled events
• immediate future events
• events that are certain or about to happen

So elkezdődik can be translated as either begins or will begin, depending on context and what sounds most natural in English.

A few helpful points:

• sz is pronounced like English s
• single s is pronounced like English sh
• stress is always on the first syllable in Hungarian words

So in visszaszámlálás:

• the sz sounds are plain s
• the spelling ssz represents a long s sound
• the stress is on the first syllable: VISsza-szám-lá-lás

For egy, the last sound gy is a soft Hungarian consonant. There is no perfect English equivalent, but an approximate feel is the dy sound in some pronunciations of during. So egy is roughly like edy, but softer.

Also, long vowels matter in Hungarian:

• ó, á, ő are long vowels
• so óra and parts of visszaszámlálás should be held slightly longer than short vowels