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Calculations of noise-assisted jump rates and diffusion coefficients for diffusion of atoms adsorbed on a metal surface are presented and discussed, in the whole range of the damping strength, and with a direct numerical integration of the Langevin equation, by two different procedures: a mean first passage time calculation and by counting jumps with a given energy criterion. The results are compared to the analytical ones by using the extension of the Kramers model by Mel'nikov and Meshkov. The role of the adiabatic potential coupling in the jump rates, jump distributions and diffusion coefficients is analysed and results are finally compared to previous experimental findings for the Na/Cu(001) system.