We watched "Warm Bodies" a couple of weeks ago. There are some scary moments, but even my "I don't like scary movies" 10 year old liked it.
You have to start slowly, and be careful about content. A little terror is okay. But sexual violence isn't. A bit of gore is okay. But realistic horror with insane characters might not be right for a 9 year old.
I think "Warm Bodies" would be a good start. And then, some classic horror, like "Night of the Living Dead" or others with no scary sexually charged scenes are good. Check on Netflix to see if the "Goosebumps" series is on there for some more age appropriate scary stuff.