Surface-electrode chips are a versatile and well-established tool for trapping and guiding charged particles. The technique, usually applied to ions, has recently been adapted for electrons using a twodimensional quadrupole guide at microwave driving frequencies. During injection into the guiding potential, the electron trajectories show a strong dependence on the phase and amplitude of fringing electric fields at the coupling entrance of the guide. Here we study the corresponding electron dynamics using a pulsed electron source with pulse durations of several 100 ps to temporally resolve fringing electric fields oscillating at the microwave drive frequency. By synchronizing the timing of the electron pulses to a certain microwave phase, we can increase the electron guiding efficiency by 50%. Furthermore, we present numerically optimized electrode structures for which the amplitude of fringing electric fields at the coupling entrance of the guide is drastically reduced. Particle-tracing simulations suggest that the optimized electrode layout allows the direct injection of electrons into the lowest-lying motional quantum states of the transverse guiding potential.