If you are willing to take a chance, just replace that capacitor with and identical one (read values off the unit). I'd give it a 70% chance that will fix it.
A starter switch is a mechanical switch that operates off the shaft. If it is on the back, you would see it. If it is deeper behind the plate, it gets a little more involved. A picture of the back of the pump motor might provide a clearer picture.
Here is my take: If.... when you power it up, the motor just hums and doesn't turn, it is likely the capacitor. If it turns a little, it might be another problem. Of course, all asumes the shaft spins freely which you said it does. If you attempt the capacitor replacement, make sure the breaker is off and short out the terminals to discharge any residual power on the capacitor terminals. Sometimes they store energy a while and you can get a jolt if you get your fingers across the terminals.