It looks like you have a 12 volt motor.
There is a switch that selects speed. 1,2,3,4,5,6,7,8,9,10,11, or 12mm/hour. (and maybe 0)
You want an Arduino to control the speed, not the switch.
It would be nice to know more about this "switch".
My guess is that this switch adds resistance to the motor to slow it down.
If you are using a micro-computer I would use a "PWM" function to reduce the voltage to the motor to slow it down.
Probably if you apply 12 volts (100% of the time) you get the 12mm/hr. If the computer were to apply 12V at 50% duty cycle then you will get a slower motor. I know someone that did this project recently. He got full speed at 12V and 1/4 speed at 50% and 1/16 at 25%. (approximately) You will need to make a lookup table.
0%=0mm/hr
10%=1mm/hr
......
......
90%=11mm/hr
98%=12mm/hr