As for the capacitor, that depends on the ballast type, if the ballast is a 120V reactor you don't need it, I don't know if HX ones need it or not. CWA/CWI ballasts always needs a capacitor.
50W HPS would be serial choke only, so the capacitor is there only to increase the power factor and generally it is not necessary for the lamp to work. It only lower the mains current, what might be important in case of long, so high resistive supply wiring is used (to reduce the voltage drop). But one 50W fixture does not draw as much to make any problem even without the capacitor...