Rare-earth anti-perovskites with oxygen are an interesting magnetic materials family at the boundary between intermetallics and oxides, they however remain largely unexplored. Here, magnetic and heat capacity investigations, as well as density functional theory (DFT) calculations, were carried out on SnOEu3. At low magnetic field (B <= 0.5 T), a Neel temperature separates antiferromagnetic and paramagnetic phases at 31 K. When applying higher magnetic field below the Neel temperature, successive transformations toward a ferromagnetic state via a number of intermediate canted magnetic structures are observed and are associated with only modest latent heat and transition entropy. High-pressure magnetic measurements confirm the stable divalent state of Eu up to 1.05 GPa. A direct magnetocaloric effect progressively increases with applied magnetic field above the Neel temperature, reaching -16 J kg(-1) K-1 for Delta B = 7 T. On the other hand, the inverse magnetocaloric effect of the field-induced transition below T-N saturates at similar to+ 5 J kg(-1) K-1. DFT calculations support magnetic instabilities observed experimentally in SnOEu 3 and reveal an unusual exchange mechanism and band topology near the Fermi level.