We present a study of superfluid vortex dynamics on general axisymmetric compact surfaces with no holes. Specifically, we develop a general method to transform conformally from the axisymmetric surface to a plane, where we use familiar methods based on the hydrodynamic stream function. Our approach shows that vortices constitute a Hamiltonian dynamical system, with their angular positions on the surface as canonical variables. The dynamical motion conserves both the total energy and the angular momentum, and it features a marked anticorrelation between the vortex speed and the local curvature of the surface.