Recent advancements utilizing large-scale video data for learning video generation models demonstrate significant potential in understanding complex physical dynamics. It suggests the feasibility of leveraging diverse robot trajectory data to develop a unified, dynamics-aware model to enhance robot manipulation. However, given the relatively small amount of available robot data, directly fitting data without considering the relationship between visual observations and actions could lead to suboptimal data utilization. To this end, we propose VidMan (Video Diffusion for Robot Manipulation), a novel framework that employs a two-stage training mechanism inspired by dual-process theory from neuroscience to enhance stability and improve data utilization efficiency. Specifically, in the first stage, VidMan is pre-trained on the Open X-Embodiment dataset (OXE) for predicting future visual trajectories in a video denoising diffusion manner, enabling the model to develop a long horizontal awareness of the environment's dynamics. In the second stage, a flexible yet effective layer-wise cross-attention adapter is introduced to transform VidMan into an efficient inverse dynamics model that predicts action modulated by the intrinsic dynamics knowledge via parameter sharing. Our VidMan framework outperforms state-of-the-art baseline models on the RLBench benchmark, achieving a 57\% and 30\% relative improvement under 10-shot and 100-shot demonstration settings, respectively, and demonstrates over 9\% precision gains on the OXE small-scale dataset. These results provide compelling evidence that world models can significantly enhance the precision of robot action prediction.