The book makes reference to the fact that Neri was "Michael's Luca". I don't remember if the film does as well, or not. Maybe in a deleted scene?

Given Neri's "special talents" I would have thought he'd be more valuable to Michael in a "Luca" role than as a Capo. Maybe that's where I got the idea that only Clemenza and Rocco were Capos after Tessio's demise.

After his introduction as a character, I think Neri was pretty much always at Michael's side. Same for Part II, except when MC was in Cuba.

I think we can safely infer from that that he was more than a Capo.

Actually, now that I think about it some more, it's kind of strange that he would be put in charge of Tessio's men as he was in the book. Why not simply promote from within? Tessio surely had a few capable underlings deserving of the spot.