This is a motor without the original controller, at this point. Find the motor, get a photo of the label, post it. It will show AC or DC operation, voltage and might even have a diagram for the wire colors. If it shows this, finding a way to operate it shouldn't be too difficult. An RF keyless entry module could be used, but it will require a DC power supply and some relays. No big deal. If it doesn't have a way to control the duration of contact closure, a timer circuit can be used. Again, no big deal. Control modules are widely available in automation and they're not necessarily expensive but we know so little about this screen that it's difficult to know what is needed.
Screens aren't rocket surgery.
If we knew the motor's brand/model/specs, a control module could probably be purchased from a manufacturer of screens. If that proves to be excessively costly, a wall-mounted switch is the easy way- momentary contact, press and hold until it's in the desired position. Done.
FWIW- some screens require a switch with more contacts because the polarity reversal happens at the switch. This one has that internally, but I have seen this when I worked for a contractor that installed Bretford screens. Had to replace a motor because someone wired the switch incorrectly.