Controlling an analog potentiometer using an Arduino
The first problem we have is that replacing the original potentiometer might be wrong thing to do. Analog audio hardware is usually nonlinear, because the volume of the sound is actually a logarithmic function. So the potentiometers might be fine tuned specifically for particular device, and impossible to acquire today since it's vintage.

I suggest a different approach. Connect the knobs to rotary encoders, or cheap linear potentiometers. That way you will receive data about what the user is doing, and what setting does he want to change. They usually are small and you might even fit an extra el-cheapo 28BYJ485V stepper motor to move the knobs to provide feedback. (It can be modified doubling it's torque)

Move all the audio potentiometers inside the case of the device and connect them directly to a stepper motor or a servo to move them only with Arduino.