Context. The thermal Sunyaev-Zeldovich (SZ) effect presents a relatively new tool for characterizing galaxy cluster merger shocks, traditionally studied through X-ray observations. Widely regarded as the “textbook example” of a cluster merger bow shock, the western, most-prominent shock front in the Bullet Cluster (1E0657-56) represents the ideal test case for such an SZ study. Aims. We aim to characterize the shock properties using deep, high-resolution interferometric SZ effect observations in combination with priors from an independent X-ray analysis. Methods. Our analysis technique relies on the reconstruction of a parametric model for the SZ signal by directly and jointly fitting data from the Atacama Large Millimeter/submillimeter Array (ALMA) and Atacama Compact Array (ACA) in Fourier space. Results. The ALMA+ACA data are primarily sensitive to the electron pressure difference across the shock front. To estimate the shock Mach number ℳ, this difference can be combined with the value for the upstream electron pressure derived from an independent Chandra X-ray analysis. In the case of instantaneous electron-ion temperature equilibration, we find ℳ = 2.08_−0.12^+0.12, in ≈ 2.4σ tension with the independent constraint from Chandra, M_X = 2.74 ± 0.25. The assumption of purely adiabatic electron temperature change across the shock leads to ℳ = 2.53_−0.25^+0.33, in better agreement with the X-ray estimate ℳ_X = 2.57 ± 0.23 derived for the same heating scenario. Conclusion. We have demonstrated that interferometric observations of the thermal SZ effect provide constraints on the properties of the shock in the Bullet Cluster that are highly complementary to X-ray observations. The combination of X-ray and SZ data yields a powerful probe of the shock properties, capable of measuring ℳ and addressing the question of electron-ion equilibration in cluster shocks. Our analysis is however limited by systematics related to the overall cluster geometry and the complexity of the post-shock gas distribution. To overcome these limitations, a simultaneous, joint-likelihood analysis of SZ and X-ray data is needed.