Oh really? I looked at Bulbapedia and it said that it couldn't learn Close Combat from the move tutor... Is that not the same person that you're talking about?
Uhh... there is a way to get Close Combat on your Staraptor. I hope you already evolved it? Ok, well go to Pastoria City and go to the house North-east of the Pokemart. The move-relearner is there, and will teach you pokemon ANY past move (including Close Combat) for a heart scale. If you don't know, heart scales are easily found while Underground.