امشب زود میخوابم چون فردا صبح باید زود به کار بروم.

Persian often uses the present tense with می‌- to talk about near-future plans, especially with time words like امشب (tonight) or فردا (tomorrow).
So امشب زود می‌خوابم is naturally understood as Tonight I’m going to sleep early / I’ll sleep early tonight, even though it’s grammatically present.

می‌ marks the imperfective aspect (habitual/ongoing/general), and it’s also commonly used for planned future actions in context.

• می‌خوابم = I sleep / I’m sleeping / I’m going to sleep (depending on context)
  Without می‌, خوابم is generally not used as a normal present tense form in modern standard Persian.

می‌خوابم = می‌ (imperfective marker) + خواب (verb stem for sleep) + -م (1st person singular ending).
Pronunciation is typically mi-xâ-bam (with خ like the German Bach sound).

چون means because and is very common in everyday Persian. You can often replace it with:

• چون‌که (because; a bit more explicit, sometimes more formal/old-fashioned)
• زیرا (که) (because; more formal/written)
• برای اینکه (in order that / because; often introduces a fuller clause and can sound more explanatory)

In this sentence, چون is the most natural, neutral choice.

Yes, Persian can switch the order of the two clauses:

• امشب زود می‌خوابم چون فردا صبح باید زود به کار بروم.
• چون فردا صبح باید زود به کار بروم، امشب زود می‌خوابم.

Both are correct. Starting with چون... can sound slightly more “structured” or written, but it’s still normal.

باید means must / have to and it commonly takes a subjunctive verb after it.
So:

• باید بروم = I have to go / I must go
  Not:
• باید می‌روم (this is generally not correct)

Here بروم is the subjunctive form of رفتن (to go).

بروم is the 1st person singular present subjunctive of رفتن (to go).
It’s built from the present stem رو plus the subjunctive prefix بِـ (often pronounced as be-), plus -م:

• بِ + رو + م → بروم

It literally means that I go, but after باید it’s understood as I have to go.

به کار رفتن is understandable and used, but in everyday speech many people more commonly say سرِ کار رفتن (to go to work / to go to the workplace).
Common alternatives:

• More colloquial: فردا صبح باید زود برم سرِ کار.
• Neutral: فردا صبح باید زود به محلِ کار بروم. (to my workplace; clearer/more explicit)

So your sentence is fine, but سرِ کار is often the most idiomatic.

Because they modify two different actions:

• امشب زود می‌خوابم = I’ll sleep early tonight.
• فردا صبح باید زود به کار بروم = Tomorrow morning I have to go to work early.

Persian often repeats an adverb like زود instead of replacing it with something like “also” or “then.”

In standard writing, it’s best as می‌خوابم with a half-space (Zero-Width Non-Joiner) between می and the verb: می‌خوابم.
In casual typing, many people write it without the half-space: میخوابم.
Both are readable, but می‌خوابم is the standard orthography.

A very common spoken version is:

• امشب زود می‌خوابم چون فردا صبح باید زود برم سرِ کار.

Changes:

• بروم → برم (colloquial shortening)
• به کار → سرِ کار (more idiomatic for “to work”)