A high-precision numerical calculation is reported for the self-energy correction to the hyperfine splitting and to the bound-electron g factor in hydrogenlike ions with low nuclear charge numbers. The binding nuclear Coulomb field is treated to all orders, and the nonperturbative remainder beyond the known Zalpha-expansion coefficients is determined. For the 3He+ ion, the nonperturbative remainder yields a contribution of -450 Hz to the normalized difference of the 1S and 2S hyperfine-structure intervals, to be compared with the experimental uncertainty of 71 Hz and with the theoretical error of 50 Hz due to other contributions. In the case of the g factor, the calculation provides the most stringent test of equivalence of the perturbative and nonperturbative approaches reported so far in the bound-state QED calculations.