ibeon ju sueobeun modu onraineuro haeyo.

이번 주 literally means “this week.”
• 이번 = “this (coming/ongoing)”
• 주 = “week”
Together, 이번 주 specifies the current week, so the speaker is talking about classes taking place during the week right now.

은 is the topic marker, indicating what the sentence is about.
• 수업은 = “As for the classes…”
Using 은 places emphasis on the topic (the classes), rather than simply stating a subject. If you used 수업이, you’d be making a neutral statement: “The classes are all online.” With 수업은, it’s more like “Regarding the classes (this week), they are all online.”

-으로 here marks means or method:
• 온라인으로 해요 = “do it by/through online.”
If you said 온라인에 해요, it would sound like you’re placing the action inside a location called “online,” which isn’t natural. 온라인에서 해요 would be like “do it at the online place,” which is also awkward. The correct way to say “by online means” is 온라인으로.

You can say 온라인 수업을 모두 해요, but the nuance shifts slightly:
• 이번 주 온라인 수업을 모두 해요.
This still means “This week, we do all (our) classes online,” but it bundles 온라인 directly with 수업 (“online classes”), rather than focusing first on “classes” as the topic. Both are correct, though the original emphasizes 수업은 (“as for this week’s classes…”).

Yes, 전부 is a formal synonym of 모두. You could say:
• 이번 주 수업은 전부 온라인으로 해요.
It means exactly the same thing (“all”), but carries a slightly more formal tone.

You can drop 모두:
• 이번 주 수업은 온라인으로 해요.
Now it simply states “The classes are (being) online this week,” which most listeners will understand as “all of them” by default. 모두 adds clarity and emphasis that there are no exceptions.