A scheme of nonreciprocal conventional phonon blockade (PB) is proposed in a spinning optomechanical resonator coupled with a two-level atom. The coherent coupling between the atom and breathing mode is mediated by the optical mode with a large detuning. Due to the Fizeau shift caused by the spinning resonator, the PB can be implemented in a nonreciprocal way. Specifically, when the spinning resonator is driven from one direction, the single-phonon (1PB) and two-phonon blockade (2PB) can be achieved by adjusting both the amplitude and frequency of the mechanical drive field, while phonon-induced tunneling (PIT) occurs when the spinning resonator is driven from the opposite direction. The PB effects are insensitive to cavity decay because of the adiabatic elimination of the optical mode, thus making the scheme more robust to the optical noise and still feasible even in a low-Q cavity. Our scheme provides a flexible method for engineering a unidirectional phonon source with external control, which is expected to be used as a chiral quantum device in quantum computing networks.