Learn Hardcore Indonesian: Saya bangun sebelum matahari terbit.

Indonesian verbs do not change form for tense. The time frame is understood from context or added time expressions. In Saya bangun sebelum matahari terbit, the word sebelum (“before”) implies that the waking up happened prior to sunrise—that places the action in the past relative to that event. If you want to be extra clear, you can add a time adverb such as tadi pagi (“this morning”):
• Saya bangun tadi pagi sebelum matahari terbit.

What does bangun mean here? Do I need to add tidur after bangun to mean wake up?

Bangun by itself can mean both wake up and get up. Adding tidur (as in bangun tidur) highlights that you are waking from sleep, but it is not required. For example:
• Saya bangun. (I woke up / I got up.)
• Saya bangun tidur. (I woke up from sleeping.)

What is the function of sebelum in this sentence?

Sebelum is a conjunction/preposition meaning before. It introduces the subordinate clause sebelum matahari terbit, which acts as a time adverbial to specify when the main action (bangun) occurred.

Why is there no article like the before matahari? How would I say “the sun” explicitly?

Indonesian does not use articles (a, an, the). Matahari can mean either sun or the sun depending on context. If you need extra emphasis or specificity, you could add a demonstrative—e.g. matahari itu (that sun)—but most of the time simply matahari is enough.

What does matahari literally mean?

It is a compound of mata meaning eye and hari meaning day, literally eye of the day. In modern usage it refers simply to the sun.

What does terbit mean? Do I need to learn ter- as a productive prefix here?

Terbit means to rise or to appear, as when the sun or moon comes up. Although Indonesian has a ter- prefix for some verbs, in terbit it is considered part of the root and not a separate affix you must attach to bit. Treat terbit as a basic verb meaning to rise.

Can I place Sebelum matahari terbit at the beginning of the sentence?

Yes. Fronting the time clause is perfectly natural and adds emphasis on the timing. For example:
• Sebelum matahari terbit, saya bangun.

Can I say Saya bangun sebelum terbit matahari instead of using matahari terbit?

Both are grammatically correct. Sebelum matahari terbit is a common fixed phrase meaning sunrise, while sebelum terbit matahari simply follows a straightforward subject-verb order in the subordinate clause. Both convey the same meaning.