Replace an Analog Pot with a digital one

You can be almost certain that the potentiometers used current are set up as voltage dividers. If you measure the three legs, they should be connected as power supply, voltage out and ground.

If this is the case, then you can just inject an analog voltage using a digitally controlled DAC if you need fast response, or use an RC filter to give an analog output from the PWM output of the Arduino if you don't.

Either way, you'll need to ensure the supply voltages match so that the highest voltage you measured on the potentiometer legs is the same as the maximum output of the DAC. By default, a standard Arduino, if you're using PWM output, will output up to 5V. Chances are the supply voltage required will be either 5V or 3.3V

I would not rely on it. While it may work there could also be some stray capacitance or other factors that also affect the reading of time and voltage value. Remember that there is a no mans land between high and low (where the analog signal would be), that theoretically could change with temperature, time or any variety of other reasons and invalidate your setup. It also relies on knowing the precise time of discharge. Keeping time like this is something that is very hard to do accurately on any microcontoller.

Take for instance you get this to work and calibrate it so that you know that it reads low when the input gets to 1.7v. But come back in a day (or even minute) later and that 1.7v threshold is now 1.6v. The calibration you did would be invalid and you would be getting bogus results.

What I would do instead is use the digital pins to read an external ADC over an SPI or other digital line. This has the benefit of being more reliable and (in most cases) more accurate as well.

So to answer your question, it is theoretically possible. It would be a tough circuit to design, you would need to disconnect the line being monitored while you charge the capacitor, but it could be done. However it most likely would not be very reliable.