Measurements of the production yields of charm baryons ($\Lambda_\mathrm{c}^+$) and charm mesons (D$^0$) in proton-lead collisions at a nucleon-nucleon center-of-mass energy of 8.16 TeV are presented. The data were collected in 2016 with the CMS experiment and correspond to an integrated luminosity of 186 nb$^{-1}$. The $\Lambda_\mathrm{c}^+$ baryon is reconstructed from the decay channel $\Lambda_\mathrm{c}^+$ $\to$ K$_\mathrm{S}^0$, while the D$^0$ meson is reconstructed via D$^0$ $\to$ K$^-$$\pi^+$. The $\Lambda_\mathrm{c}^+$ baryon and D$^0$ meson yields are extracted in several charged-particle multiplicity classes. No strong multiplicity dependence of the $\Lambda_\mathrm{c}^+$-to-D$^0$ yield ratio is observed, unlike the observed strange baryon to strange meson yield ratio of $\Lambda/\bar{\Lambda}$ to K$_\mathrm{S}^0$, which shows a strong multiplicity dependence. This observation indicates different mechanisms for the multiplicity evolution of hadronization processes for charm and strange quarks and provides new constraints to the understanding of heavy flavor production and collectivity in small collision systems.