This is, coincidentally, exactly what we're doing in Actuator Design and Integration right now. If you happen to know a student in the class they can give you some good advice on this.
Here are some quick thoughts, but not a thorough answer:
I agree that you need to support the box properly, and then simply rotate it via something like a belt with a motor on the side. You'll need to attach a strong shaft to the back of the box and support it inside of something that allows it to rotate but prevents it from sliding out. That would be a bushing or roller bearing.
I would be tempted to build a box behind your box to house the rotation supports and motor. Then you can hang that entire assembly on the wall just like any other object.
While not designed for vertical use, you might get away with using a medium size one of these:
Hope that helps get you started,