Magnetic properties of a one-dimensional multicomponent quantum-spin system with an excitation gap are studied based on the integrable spin model introduced by Bariev et al. [Z. Phys. B 96, 395 (1995)]. By exactly computing the magnetization, we show how a characteristic structure with plateaus and cusps appears in the magnetization process. To study the low-energy dynamics of the system, we apply a finite-size scaling analysis to the excitation spectrum, and thereby evaluate the power-law exponent as well as the enhancement factor for the low-temperature NMR relaxation rate 1/T-1. We discuss the critical properties of 1/T-1 around plateaus and cusps in the magnetization curve.