We report on a theoretical and experimental study of state-selective differential single-electron transfer cross sections between Na$^{+}$ ions and Rb(5s,5p) atoms at collision energies of 2, 5, and 7 keV. A two-center multichannel semiclassical impact parameter close-coupling method with straight-line trajectories was used to obtain single-electron capture amplitudes. By combining with the eikonal approximation, we calculated the angular differential cross sections. These results are compared to the experimental data obtained with Rb targets cooled in a magnetic optical trap. It is shown that there is generally a good agreement between the present calculations and the experiments. In spite of the higher resolution offered from the cold target, the rapid oscillations in the differential cross sections are not resolved by the experiments.