Well - there's no way to answer the motor question without all of the speed, acceleration and mass data. Yes - a stepper will do this. Might be a big one, might be a small one. Might be a gear-reduced one.
How to make the motion depends on your design and space constraints. Have you though of using a rack and pinion? Or a ballscrew? Lots of ways but it depends on a lot of things and we do not have anywhere near enough info to make a recommendation.
Scott
Consistency is a good thing....unless you're consistently an idiot.