I would have thought that the simplest and easiest (to build and operate) solution would be to use an ordinary household mains timer to define when the door should be open and closed. You can use that to switch a relay which toggles the door to either go up until it hits a limit switch, or goes down until it hits a limit switch. The hardest part would be assembling the mechanism to move the door, and that's a non-Arduino problem which will be equally hard whichever solution you choose. (A cheap and easy option would be to get an electric window lifter from a car breakers and set it up to pull a cord which raises the door; gravity would lower the door for you, so no worries about crushing anything.)