Cadmium sulfide is one of the cutting-edge materials of current optoelectronic technology. Although many theoretical works are presented the for pressure-dependent elastic and related properties of the zinc blende crystal structure of cadmium sulfide, there is still some scarcity for the elastic, mechanical, and phonon behavior of the wurtzitic phase of this important material under pressure. In contrast to former theoretical works and methods used in literature, we report for the first time the application of a recent shell model-based interatomic potential via geometry optimization computations. Elastic constants, elastic wave velocities, bulk, Young, and shear moduli, as well as the phonon behavior of wurtzite cadmium sulfide (w-CdS) were investigated from ground state to pressures up to 5 GPa. Calculated results of these elastic parameters for the ground state of w-CdS are approximately the same as in earlier experiments and better than published theoretical data. Our results for w-CdS under pressure are also reasonable with previous calculations, and similar pressure trends were found for the mentioned quantities of w-CdS.