I'm imagining this as similar to desk fans which are common here, with big clunky buttons to select different fan speeds and a mechanism to stop more than one speed from being selected at once. You'd need to replace all of that with a set of relays controlled by the Arduino. If you wanted to allow direct manual control as well as remote control, you'd then need to add a corresponding set of momentary switches that the Arduino could read so that you could drive it manually. It all seems doable, although the only way the project makes sense to me is as a learning exercise.