Here is another method just like an old-fashioned mechanical radio dial. A large continuous loop of string goes around 3 pulleys: (a) one on the motor shaft, and (b) one at each extreme end of travel for your device. Attach your device to the string. Spin the motor in either direction to make your device move up or down.
This method easily gets out of calibration. It helps if you put spring tension on the string, and make the pulleys spin freely.
This is okay for short distances. It requires attaching the screw to the motor shaft. The nut must be free to travel the entire length of the screw, but you must also attach it with a ring to a stationary shaft so it does not spin around. Your device is carried up and down as the nut travels up and down the threaded shaft. You need to build in enough flexibility in the joints, so that there is no binding and seizing during operation.