The motors in these fans are single phase, with a single winding that is reversed each time a magnet passes the Hall sensor (positioned between the poles). Current through the motor is then reversed, either by reversing the connections, or having two windings that are alternately energized.
To start up, the poles are asymmetric, and the magnets align themselves with the fat ends of the pole pieces when there is no phase current. When power is applied, magnets align with the full width of the pole pieces, and the rotor's momentum carries it round as the phase current is reversed, and the magnets are then attracted to the next pole.
The issue is that without a sudden application of power, the rotor will not have enough momentum to carry on past the aligned position, where the torque generated by the motor is zero. Gradually raising the voltage will very slowly move the rotor into alignment.
You need some means of switching the power to the motor on as the capacitor passes the minimum voltage for the motor. A simple, though not very efficient way is to use a relay, coil wired across the solar cell, that will close contacts at a defined voltage, and has a fairly wide hysteresis (drop out voltage is usually quite a lot lower than the pull-in) so that it won't disconnect as the motor draws startup current. A simple circuit with a comparator switching the motor using a FET would be more efficient.