Cowpox is caused by a DNA virus known as the cowpox virus (CPXV) belonging to the Orthopoxvirus genus in the family Poxviridae. Cowpox is a zoonotic disease with the broadest host range among the known poxviruses. The natural reservoir hosts of CPXV are wild rodents. Recently, the cases of orthopoxviral infections have been increasing worldwide, and cowpox is considered the most common orthopoxviral infection in Europe. Cowpox is often a self-limiting disease, although cidofovir or anti-vaccinia gammaglobulin can be used in severe and disseminated cases of human cowpox. In this computational study, a molecular docking analysis of thymine- and arabinofuranosyl-thymine-related structures (1–21) on two cowpox-encoded proteins was performed with respect to the cidofovir standard and a 3D ligand-based pharmacophore model was generated. Three chemical structures (PubChem IDs: 123370001, 154137224, and 90413364) were identified as potential candidates for anti-cowpox agents. Further studies combining in vitro and in silico molecular dynamics simulations to test the stability of these promising compounds could effectively improve the future design of cowpox virus inhibitors, as molecular docking studies are not sufficient to consider a ligand a potential drug.