Optimizing intrathecal drug delivery procedures requires a deeper understanding of flow and transport in the spinal canal. Numerical modeling of drug dispersion is challenging because of the strong separation of time scales: dispersion occurs over approximately one hour, whereas cerebrospinal fluid pulsations driven by cardiac motion occur on a time scale of about one second. Patient-specific predictions in clinical settings therefore call for simplified descriptions that focus on dispersion time scales while bypassing the rapid concentration oscillations induced by cyclic motion. We show how asymptotic methods that exploit this separation of time scales can be used to derive a reduced transport equation in which convective transport driven by the mean Lagrangian drift governs drug dispersion. The model is validated through comparisons with MRI-informed direct numerical simulations of drug dispersion in a cervical spinal canal geometry that includes nerve rootlets and denticulate ligaments. These comparisons demonstrate that the reduced model accurately captures drug transport while enabling dispersion predictions at a fraction of the computational cost required by direct numerical simulations.