Learn Italian: Non che voglia sembrare ingrato, ma l'orario si può cambiare? - Not that I want to seem ungrateful, but can the schedule be changed?

The phrase non che (not that) introduces a rejected cause. You are bringing up a potential reason just to deny it. In Italian, this requires the subjunctive mood because it represents a hypothetical situation. The actual point you want to make follows ma (but) and uses the indicative mood, as seen with si può.

Since the present subjunctive forms for io, tu, and lui/lei are all voglia, how do we know the subject is I?

Context usually makes it clear, but in this specific sentence, we also have the adjective ingrato (ungrateful). Since the speaker is expressing their own concern about how they might appear, we understand the hidden subject is io. If the speaker were a woman, she would need to say ingrata.

Why does the sentence say l'orario si può cambiare instead of using a standard passive like può essere cambiato?

This uses the passive si (si passivante), which is much more common in everyday spoken Italian than the traditional passive voice. When combining the passive si with a modal verb like potere (can), the si usually goes before the conjugated modal verb, while the main verb (cambiare) remains in the infinitive.

Does l'orario have to go at the beginning of the second clause?