What you're trying to make sounds like a drill press and an indexing table. I think the easiest and most reliable approach would be to use a conventional drill press - or make your own to a similar design.
I'm envisaging something reasonably substantial and I doubt that an ordinary hobby servo would be powerful enough to operate it, but I'm sure you could find a geared motor which would cope with it. For simplicity I'd be inclined to use limit switches to control the vertical travel, but if you want to make it more general then a stepper motor/servo would give arbitrary depth control.