Two of the most pressing challenges in cosmology are the persistent discrepancy in measurements of the Hubble constant, referred to as the Hubble tension, and the deficit of baryons in the local Universe, known as the missing baryon problem. Fast radio bursts (FRBs) provide a unique probe of both the Hubble constant H0 and the cosmic baryon density Ωb. However, constraints from FRBs alone suffer from a severe H0–Ωb degeneracy that prevents them from resolving either problem. We show that this degeneracy can be broken by combining FRBs with other emerging probes whose degeneracy directions differ in the H0–Ωb plane. Specifically, we quantify three multi-messenger approaches: FRBs paired with gravitational wave (GW) standard sirens, strong gravitational lensing (SGL) time delays, and 21 cm intensity mapping (IM) surveys. The combinations FRB+GW, FRB+SGL, and FRB+21 cm IM each deliver simultaneous constraints on H0 and Ωb better than (1%, 1.5%) in the ΛCDM model, and when dynamical dark energy is introduced, the constraining precision degrades gracefully as model complexity increases. Furthermore, within a model-independent framework, both FRB+GW and FRB+SGL constrain H0 and Ωb to precisions better than (1.5%, 3%). These precision levels are based on nominal observational expectations and would improve significantly under optimistic observational scenarios.