Charm quark production and its hadronization in ep and eA collisions at the future Electron-Ion Collider in China (EicC) will help us understand the quark/gluon fragmentation processes and the hadronization mechanisms in the nuclear medium, especially within a poorly constrained kinematic region ($x<0.1$). In this paper, we report a study on the production of charmed hadrons, $D^0$ and $\Lambda_c^+$, reconstructed with a dedicated GEANT4 simulation of vertex$\,\&\,$tracking detectors designed for EicC. The $\Lambda_c^+$/$D^0$ ratios as functions of multiplicity and $p_T$, as well as the $D^0$ double ratio are presented with projected statistical precision.