We perform a numerical study of interacting one-dimensional Hubbard rings with a single impurity potential and pierced by a magnetic flux. Our calculations are carried out at the level of current lattice density functional theory (CLDFT) for the Hubbard model and compared to known results obtained in the thermodynamical limit from the Bethe ansatz. In particular, we investigate the effects of disorder and Coulomb interaction on the persistent current (PC) and the Drude weight. It is found that CLDFT is able to accurately describe qualitative and quantitative features of these ground state properties in the presence of disorder and electronic interaction. When the impurity potential is switched off, the CLDFT approach describes well the velocity of the Luttinger liquid excitations as a function of both interaction strength and electron filling. Then, when the impurity scattering potential is finite, we find the PC to vanish as ${{L}^{-{{\alpha}_{\text{B}}}-1}}$ for large L and independent on the strength of the scattering potential, in good agreement with Luttinger liquid theory.