Learn Turkish: Sınavı geçmek için ders çalışıyorum.

You are right that için usually means "for". However, when you put için immediately after a dictionary-form verb (an infinitive ending in -mak or -mek), the combination means "in order to" or just "to". So, geçmek için translates to "in order to pass".

Why is it sınavı and not just sınav?

The -ı on the end of sınav (exam) is the accusative case ending. We use this ending to point out a specific direct object. The speaker isn't trying to pass just any test; they want to pass "the" exam. Because it's a specific object receiving the action of the verb geçmek (to pass), it becomes sınavı.

If çalışıyorum means I am studying, why is the word ders in there?

By itself, the verb çalışmak usually means "to work" (like at a job) or "to try". To specifically say "to study", Turkish uses the fixed phrase ders çalışmak, which literally translates to "to work a lesson". So, ders çalışıyorum is how you say "I am studying".

Why does the in order to pass part come at the beginning of the sentence?

In standard Turkish grammar, the main verb almost always goes at the very end of the sentence. The phrase sınavı geçmek için (in order to pass the exam) gives the reason for the action, so it is placed before the main verb phrase ders çalışıyorum (I am studying).