Optical bistability is a crucial nonlinear optical phenomena that is integral to all-optical switching, photonic logic systems, and quantum information processing. This paper systematically examines the cooperative regulation mechanism of magnon bistability and optical bistability in a hybrid atom-cavity opto-magneto-mechanical system. This system comprises a microwave cavity, a magnon mode supported by yttrium iron garnet (YIG) crystal, a mechanical phonon mode, dual coupled optical cavities, and a two-level atomic ensemble. Multi-mode coupling is achieved through magnetostrictive interaction, optical radiation pressure, and atomic dipole-cavity coupling, addressing the issues of limited tuning freedom and weak nonlinear response in traditional optomechanical systems. Utilizing the whole quantum Hamiltonian of the system, we formulate the quantum Langevin equations incorporating dissipation and quantum noise factors, and extract the steady-state analytical solutions for magnon population and optical cavity photon number under robust driving circumstances. Numerical simulations are conducted using experimentally viable parameters to examine the influence of coupling strengths, mode detunings, and dissipation coefficients on the threshold, hysteresis breadth, and steady-state amplitude of magnon and optical bistability. The findings indicate that both magnon and optical bistability can be accurately controlled by modifying essential system parameters. The incorporation of an atomic ensemble introduces an additional nonlinear interaction pathway, thereby substantially improving the tunability and stability of the hybrid system. Furthermore, the sudden state transition at the critical driving point offers a robust physical foundation for the development of high-speed magneto-optical switching devices. This study elucidates the multi-mode coupling synergistic mechanism underlying bistable responses in atom-cavity opto-magneto-mechanical systems, establishes comprehensive quantitative regulation principles for bistable characteristics, enhances the foundational theory of hybrid opto-magneto-mechanical systems, and offers robust theoretical support for the design and development of tunable, low-threshold, and high-stability magneto-optical devices. This research holds significant theoretical value and extensive engineering application potential in quantum information processing, microwave-optical signal transduction, and high-precision quantum sensing.