What is the easiest and most inexpensive arduino solution for that?
But then I'm still not sure how to get it to move in a straight line
retrolefty: that sounds complex and expensive. Is there any kit for that that I can just plug the rod into and then control from arduino?
Another method is to use a stepper motor attached to a thread rod, and then have a threaded follower travel up and down the rod as the motor rotates in either direction.
Please believe me when I say that I can't think of any more details that are important and would not only distract and that I'd rather not into
(MacGyver also has no shoestring, chewing gum or paper clips at hand so there are no quick workarounds. He needs Arduino!)
You could make a very small scissor lift, a DIY linear actuator (I've seen a small one on youtube using a servo motor), a push/pull cable like the ones in bicycle controls, telescoping rod sections, ada, yada, yada.