אוקיי, המשחק מתחיל.

A natural pronunciation is:

o-KEI, ha-mis-KHAK mat-KHIL

A word-by-word guide:

• אוקיי = o-KEI
• המשחק = ha-mis-KHAK
• מתחיל = mat-KHIL

The kh sound represents Hebrew ח, a throaty sound like the ch in German Bach or Scottish loch.

Yes, it is very common in modern Hebrew, but it is a borrowed word from English okay.

Hebrew speakers use אוקיי much like English speakers use okay:

• to start something
• to react
• to move the conversation forward

In this sentence, it works like Okay, ... at the beginning of an announcement.

You may also see other spellings such as אוקי.

In Hebrew, the definite article the is usually a prefix: ה־

So:

• משחק = game
• המשחק = the game

This is different from English, where the is a separate word. In Hebrew, it usually sticks directly to the noun.

מתחיל is the present-tense masculine singular form of the verb להתחיל, which means to start or to begin.

So:

• להתחיל = to start
• מתחיל = starts / is starting / beginning

You use להתחיל when you mean the infinitive, like to start.

You use מתחיל when the action is happening in the sentence with a subject, as in:

• המשחק מתחיל = the game starts / is starting

Because in Hebrew, the verb to be is usually omitted in the present tense.

So Hebrew often says something more like:

• the game starting

But in natural English, that becomes:

• the game is starting
• the game starts

This is very normal in Hebrew. You do not need a separate word for is here.

It can mean any of those, depending on context.

Hebrew present tense often covers both:

• simple present = starts / begins
• progressive present = is starting

So המשחק מתחיל could be understood as:

• the game starts
• the game is starting
• the game begins

In many real situations, English would choose is starting if it is happening right now.

Because Hebrew verbs in the present tense agree with the subject in gender and number.

The subject here is המשחק = the game, and משחק is a masculine singular noun, so the verb is also masculine singular:

• המשחק מתחיל

Compare:

• ההצגה מתחילה = the show is starting
  feminine singular

• המשחקים מתחילים = the games are starting
  masculine plural

• ההצגות מתחילות = the shows are starting
  feminine plural

You might hear unusual word orders in Hebrew, but המשחק מתחיל is the most neutral and natural way to say it in everyday speech.

Putting the verb first, as in מתחיל המשחק, sounds more marked, more dramatic, or more literary. It is not the default choice for a simple everyday sentence like this one.

So for normal speech, המשחק מתחיל is the best pattern to learn.

Yes. אוקיי is optional.

• אוקיי, המשחק מתחיל. = Okay, the game is starting.
• המשחק מתחיל. = The game is starting.

Adding אוקיי gives the sentence a more conversational or announcer-like tone, as if someone is getting things going.