Most Americans would say simply: "I'm about to finish graduate school." We know you're 'graduating' if you're 'finishing'

I disagree with the others. I suggest you don't need the last part of the phrase. I would write "I'm about to graduate". Because where else whould you graduate from? You may find it useful to look up "tautology" in a dictionary. Spinal Tap sang a song "Tonight I am going to rock your world tonight." as you can see this has one too many words in it.

Hi Keita, You can say this. Grammatically, there's nothing wrong with it, and as Erin mentioned, the two are pronounced differently (at least in American English). When used as a verb, the second "a" in "graduate" is pronounced as a long "a" sound /ˈgrædʒ uˌeɪt/, and when used as an adjective, the second "a" becomes a short "i" sound: /ˈgrædʒ u ɪt/.

Both are fine to use. I think in this case it helps that, at least in my accent, I hear the verb graduate as gradu(A)te and the adjective graduate as gradu(it). The difference in pronunciation makes it sound less repetitive to me.

I feel like it is ok. Because first graduate acts as a verb and second one act as a noun as I understand.