Neri does come and go pretty strangely in Part III -- I will agree with that. And someone does bring up an excellent question about Neri still being around -- especially if he reminds Michael of killing Fredo.

I think that up until Part III, Michael has only started to regret killing Fredo, so he wouldn't have done away with Neri quite yet. By the time Mary is dead on the opera steps, Michael has realized the entirity of his sins and at that point would be finished with everyone he ever did business with.