Coexistence of nontrivial topology and flat electronic bands provides a fertile platform for correlated quantum states. The square-octagon lattice hosts Dirac nodes and flat bands at half-filling, yet the effects of intrinsic spin-orbit coupling (SOC) and staggered magnetic flux on its electronic and topological properties remain largely unexplored. Here, using tight-binding models incorporating SOC and staggered magnetic flux, we uncover a rich topological phase diagram in this lattice, comprising a quantum spin Hall phase with spin Chern number Cₛ=1, crossovers to quantum anomalous Hall phases with C=1 and C=2, and higher-order topological insulator phases with quantized quadrupolar corner charges. The initially dispersionless flat bands evolve into quasi-flat topological bands with nearly uniform quantum geometry and large flatness ratios, making them promising candidates for fractional Chern insulator states. We further identify realistic materials, including octagraphene, transition-metal dichalcogenides, synthetic MoSi₂N₄, and magnetic α-MnO₂, that may realize these tunable topological phases intertwined with flat-band physics, opening new opportunities for correlated topological matter.