An Nd:YAG/Cr4+:YAG passively Q-switched (PQS) laser in a near-hemispherical cavity is exploited to generate high-order structured pulsed fields. Under tightly focused on-axis pumping, radial-order Laguerre-Gaussian (LG) modes with controllable mode orders by the input pump power are realized to exhibit quite stable temporal behavior. The pulse repetition rates of radial-LG modes can reach up to 78 kHz with an average output power of 0.57 W and peak power beyond 300 W under a 5-W pump level. Furthermore, by introducing 1D off-axis pumping into the PQS laser, various structured pulsed fields with transverse morphologies as high-order Ince-Gaussian (IG) modes are further created. With clean and well-defined beam structures, the IG pulsed fields can be nicely reconstructed by the resonant modes of the inhomogeneous Helmholtz equation for spherical cavities. More importantly, these high-order PQS IG modes reveal highly regular pulse trains with the maximum pulse repetition rate beyond 20 kHz and overall peak power higher than 1.5 kW.