In the design and fabrication of molds in the deep drawing process, the gap between the punch and the mold is extremely small. Obtaining a product of the correct size and shape necessitates the control of certain variables in the process, consideration of the problems of springback after load removal and the occurrence of cracks in the drawing process, and the estimation of the punch load. In this study, we focused on copper sheets and considered the influence of the scale effect and fillet radius of a micro square hole on thin sheets on the basis of an updated Lagrangian formulation (ULF) and finite element analysis. Sheet behavior was simulated using a micro-elastoplastic material model, the performance of which was compared with that of models involving conventional materials. Subsequently, the Dynaform LS-DYNA solver was used for simulation analysis, and pre- and post-processing were carried out to obtain the material deformation history, as well as to determine the thickness change distribution and material stress and prepare strain distribution maps. It was found that the scale effect of the sheet thickness influences the relation between the punch load and stroke, the distribution of thickness, the distribution of stress and strain, the maximum diameter of the flange hole, and the maximum flange height. Finally, the simulation results were compared with experimental results to confirm the accuracy of 3D finite element analysis of the elastoplastic deformation. The results show the influence of the fillet radius of the inner hole (Br) for copper sheets on the drawing process: the punch load increases with increasing Br, but the minimum thickness of the formed flange decreases as Br increases. The maximum principal stress/strain and the flange height increase as Br increases. The findings serve as a valuable reference for the design and processing of micro deep drawing.