You can pick up a linear actuator on E-Bay or Amazon for probably around $50, maybe less. I bought mine a few years ago, so I don't know current pricing.
Then you just need a switch and some wire to make it work. A heavy duty double pole double throw, monetary contact, center off is all you need. The actuator will probably come with mounting brackets, which will most likely work if you take the time to find the right placement. I think my actuator has about 8" of travel, but you don't need that much, 4" is more like it. Look for an actuator with built in adjustable limit switches.