What kind of component lets me measure distance in a small space?
My suggestion is that you forget about trying to figure out where the needle is at and build additional circuitry that interfaces with the radios tuned circuit.

You could select the AM band which tunes the internal oscillator from about 500kHz to 1.7Mhz.  If you got your hands on the schematics for this radio you could figure out where to put your "tapping" point.  The problem is not to load the circuit in a way that would alter the frequency, but that's doable.

It could even be as simple as reading the value of the variable capacitor that the dial is altering.

Doing it this way, you are less likely to deface a beautiful old radio and you'll know a bit about how radios work when it's done.