Rather than emulating the POT, a more practical approach is to generate the control signal directly.
For this you will need a digital-to-analog (DAC) converter. In its simplest form a DAC can be built with a series resistor and a capacitor (RC element). The output from the DAC will then take the place of the mechanical wiper (the POT center pin). To set desired voltage output, you can use a digital Arduino pin to either charge (output high) or discharge (output low) the capacitor based on feedback to an analog input (ADC). Analog feedback allows you set and track the desired voltage output with high precision and works well with a simple RC element.