The answer is "Coincidentally, my oldest sibling, Nate, just turned nineteen."
Why? Well, if we look at A: "I" or the speaker in question is the subject talking about and or completing an action.
C: "My brother Nate" is already the subject, because Nate is the first person being spoken about.
Meanwhile in B: "Nate" Is not the subject, Nate is the name that describes, or specifies who the subject is.
So you got older sibling(Subject). Who is your older sibling? Nate,