To do this on a budget, I would start at the local auto junk yard. Modern cars are full of small 12V DC gear motors; think windshield wiper motor, mirror movers, window winders, door locks, seat movers, etc. Something like a seat actuator, or maybe the wiper motor might be a place to start.
You will need "limit switches" to stop the motor when it reaches the appropriate position. Figure what gear motor might work for you, and write back and we'll give a schematic that shows how to preselect the position, and have the motor stop automatically in the up or down position.