Optomechanics, where photons (e.g. optical, microwave) are coupled to mechanical motion, provide the tools to control mechanical motion near the fundamental quantum limits. Reaching single-photon strong coupling would allow to prepare the mechanical resonator in non-Gaussian quantum states. Yet, this regime remains challenging to achieve with massive resonators due to the small optomechanical couplings. Here we demonstrate a novel approach where a massive mechanical resonator is magnetically coupled to a microwave cavity. By improving the coupling by one order of magnitude over current microwave optomechanical systems, we achieve single-photon strong cooperativity, an important intermediate step to reach single-photon strong coupling. Such strong interaction allows for cooling the mechanical resonator with on average a single photon in the microwave cavity. Beyond tests for quantum foundations, our approach is also well suited as a quantum sensor or a microwave to optical transducer.