Hi,
There are a couple of approaches that may work:
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Mechanically couple a standard servo with, say, 120 degree rotation with a 3:1 ratio, using gears or two small spools/drums and thin strong string like braided fishing line. Then each 20 degrees of servo movement gives 60 degrees on your shifter. calibrate values for each position and put in your arduino software.
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use a continuous rotation servo directly coupled to the shifter. Make a disk with holes 60 degrees apart and use an opto-interrupter or IR LED and Phototransistor to detect when in position. Count and keep track of position. Add microswitch to disconnect servo power if a fault causes rotation to unallowed location.
Have fun, and please let us know what works out!