Learn German: Ich muss arbeiten. Es ist gleich neun.

You are right that gleich often means "right away" or "in a moment" (like "I will do it right away"). However, when telling time, es ist gleich... is a very common set phrase that means it is "almost" or "just about to be" that hour.

Why is the word Uhr missing after neun?

Just like in English, you can leave out the word for "o'clock" (Uhr) in casual conversation when the context is clear. Saying Es ist gleich neun is very natural, just like saying "It is almost nine."

Could I say kurz vor instead of gleich in this sentence?

Yes, absolutely! Es ist kurz vor neun (It is shortly before nine) means the exact same thing. German speakers use both phrases interchangeably to say that an hour is just a few minutes away.

Why is arbeiten at the end of the first sentence? Shouldn't it be "Ich muss arbeiten" word-for-word?

It actually is word-for-word in this specific, short sentence! But as a quick refresher on German sentence structure: when you use a "helping" verb like müssen (must / have to), the conjugated verb (muss) takes the second position, and the main action verb gets pushed to the very end of the clause in its dictionary form (arbeiten). If you added more words, like "today" (heute), it would become Ich muss heute arbeiten.