A-Posteriori Error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewiseconstants $u_0$ and cellwise $q$-order reconstructed functions $R^0_q u_0$. Using standard duality arguments and the known theory for the discontinuous Galerkin FEM, we construct  exact error representation formulas for derived functionals that are tailored to the class of high order Godunov finite volume methods with data reconstruction, $R^0_q u_0$. We then consider computable error estimates that exploit the structure of Godunov finite volume metIssues common to the discontinuous Galerkin method and Godunov finite volume method such as the treatment of nonlinearity, postprocessing of dual data, and extension to systems of conservation laws are considered. Numerical results for advection, diffusion, and advection-diffusion equations are presented to validate the analysis.