In this paper, we design and analyze staggered discontinuous Galerkin methods of arbitrary polynomial orders for the stationary Navier-Stokes equations on polygonal meshes. The exact divergence-free condition for the velocity is satisfied without any postprocessing. The resulting method is pressure-robust so that the pressure approximation does not influence the velocity approximation. A new nonlinear convective term that earning non-negativity is proposed. The optimal convergence estimates for all the variables in $L^2$ norm are proved. Also, assuming that the rotational part of the forcing term is small enough, we are able to prove that the velocity error is independent of the Reynolds number and of the pressure. Furthermore, superconvergence can be achieved for velocity under a suitable projection. Numerical experiments are provided to validate the theoretical findings and demonstrate the performances of the proposed method.