In earlier work [H. Liu and Z. Wang, J. Comput. Phys., 328(2017)], an arbitrary high-order conservative and energy-dissipative direct discontinuous Galerkin (DDG) scheme was developed. Although this scheme enforced solution positivity using cell averages as reference values, it lacked a theoretical guarantee for the positivity of those cell averages. In this study, we develop a novel arbitrary high-order DDG method with rigorously proven positivity-preserving properties. Specifically, the positivity of the cell averages is ensured through a modified numerical flux in combination with forward Euler time discretization. To achieve point-wise positivity of ion concentrations, we introduce a hybrid algorithm that integrates a positivity-preserving limiter. The proposed method is further extended to higher-dimensional problems with rectangular meshes. Numerical results confirm the scheme's high-order accuracy, guaranteed positivity preservation, and consistent discrete energy dissipation.