You could use 12 photo transistor / photodiode pairs and have a small "arm" or "flap" on the moving part that goes between the diode and transistor and use that to report the position. You would of coruse have to keep track of whitch direction you are moving somehow.
In the printer you scavenged the motor mechanism from you can probably find at least one of these photo transistor / photodiode pairs. They are frequently used in printers to detect when the printhead reaches "the end" of it's allowe travel. They are usually encased in a smal "U shaped" black plastic package.
Another good source for photo transistor / photodiode pairs are computer mice (old ones from before optical mice) they usually have two pairs.
I think each photo transistor / photodiode pair could be hooked up to Arduino like any other switch, but i have not tried that.
EDIT:
I think (like Grumpy suggested) a stepper motor would be a better solution for this task, it's not exactly trivial to control a DC motor with the precision you need. Some printers have stepper motors for printhead movement, but most have a DC motor. On the other hand, most printers use a stepper motor to feedthe paper, maybe you could hack the whole assembly to use the steppermotor in stead of the DC motor ? (probably mechanically difficult)