Electrophoretic and electroosmotic motion of a charged spherical particle within a cylindrical pore filled with a Debye-Bueche-Brinkman (DBB) polymeric solution is investigated theoretically, which is of high relevance in capillary electrophoresis as well as micro- and nanofluidic applications involving polymeric solutions in a micro- or nanopore. The DBB model describes the rheological response of a polymeric solution with a linear polymer dissolved in a homogeneous solvent. It is a well-known non-Newtonian model in liquid physics based on rigorous theoretical derivations. By Debye and Bueche, corresponding governing fundamental electrokinetic equations are solved numerically with a patched pseudo-spectral method based on Chebyshev polynomials. We found that the double-layer polarization effect reduces the particle mobility severely when the Debye parameter, κa, is around unity, especially in narrow pores. This is attributed to the extra confinement effect from the nearby wall, which tends to sweep the predominant counterions within the double layer to the wake of the moving particle, resulting in a motion-deterring induced electric field. The electrophoretic mobility in a polymer solution is smaller than that in an aqueous electrolyte solution in general as a result of the much stronger viscous drag effect in a polymer solution. Moreover, electroosmotic flow (EOF) as a result of a charged pore wall is found to exhibit a highly non-Newtonian behavior. Unlike the corresponding plug-like flow for a Newtonian solution, an axisymmetric flow with a large local maximum in the velocity profile in the region near the pore wall is observed. This radial-varying velocity profile offers a potential extra separation mechanism, which favors the elution of smaller particles in general. The results obtained here provide fundamental understandings and insights of the electrophoresis and electroosmosis phenomena in a cylindrical pore filled with polymeric solution.