You could modify the wiring of the receptacle so that it is not switched, but always on. then put in a remote controllable switch... X-10, Crestron, Lutron, whatever product you currently use and compatible with the control system you have installed.

In any event I'm not sure that hard powering off the projector is really the best method of approaching lift mounted projectors. The cool down process is there for a reason, and it is best to leave it up to the projector to determine when it is ready to turn off, rather than hard powering it off, not to mention possible issues with power supply life. If all else fails, there's always the Clapper.