Hydrodynamical instabilities may either spin-up or down the pulsar formed in the collapse of a rotating massive star. Using numerical simulations of an idealized setup, we investigate the impact of progenitor rotation on the shock dynamics. The amplitude of the spiral mode of the Standing Accretion Shock Instability (SASI) increases with rotation only if the shock to the neutron star radii ratio is large enough. At large rotation rates, a corotation instability, also known as low-T/W, develops and leads to a more vigorous spiral mode. We estimate the range of stellar rotation rates for which pulsars are spun up or down by SASI. In the presence of a corotation instability, the spin-down efficiency is less than 30%. Given observational data, these results suggest that rapid progenitor rotation might not play a significant hydrodynamical role in the majority of core-collapse supernovae.