One of the most important problems of metamaterials and metasurfaces research is the derivation and the analysis of the effective parameters. They allow to examine the structure without singling out each element and it is the significant advantage for practical use. Recently, it has been shown that in virtue of a subwavelength thickness metasurfaces can be described within an effective conductivity approach. Such an effective surface conductivity describes the properties of a metasurface in the far-field as well as in the near-field. We derive and analyze the effective surface conductivity of a plasmonic resonant anisotropic metasurface theoretically and numerically. With the help of obtained effective conductivity we study the near-field properties of this metasurface, in particular, the equal frequency contours of surface waves. We show the topological transition from elliptical to hyperbolic-like dispersion regime for the surface waves on a hyperbolic metasurface. Finally, we study the influence of spatial dispersion on the eigenmodes spectrum and analyze the hyperbolic regime of a metasurface with strong spatial dispersion.