Cosmic rays are believed to partially control the physical and chemical evolution of molecular clouds. In fact, the impact of cosmic rays on these star-forming regions could be approximately quantified via the cosmic-ray induced ionization rate $\zeta({\rm H}_2)$. The aim of this work is to provide estimates for this quantity in the prototypical starburst nucleus of NGC 253 using non-thermal emissions (X-ray, GeV and TeV gamma-ray) from this objects. To this end, we employ a cosmic-ray transport model to fit data for non-thermal emissions and derive the cosmic-ray spectra in this system. We then adopt these cosmic-ray spectra to evaluate the ionization rates and find the values of $\zeta({\rm H}_2)$ to be around $4\times 10^{-14}$ s$^{-1}$ which is about 2 to 3 orders of magnitude higher than the typical values found in molecular clouds of the Milky Way. Such a high value of $\zeta({\rm H}_2)$ is mostly due to the fact that these nuclei, with typical sizes of a few hundred parsecs, have relatively high rates of supernova explosions which are comparable or even higher than that of the entire Milky Way. Interestingly, the ionization rates have been inferred (using molecular line observations) to be around $10^{-13}$, and in some extreme cases reaching $10^{-12}$ s$^{-1}$, for several clouds in the central molecular zone of NGC 253 which is higher than the values derived from our fit to the non-thermal emissions. We will discuss in more detail potential explanations for this discrepancy. The framework presented in this work, however, clearly illustrates the potential of non-thermal emissions as a tool to better quantify the impact of cosmic-ray in starburst environments and, in the future, might help us to gain more insights into the important role of cosmic rays as regulators of star-forming regions.