The Belle II experiment at the asymmetric $e^+e^-$ collider, SuperKEKB, aims to record 50 ab$^{-1}$ of data over the next decade, a factor of 50 more than Belle. During the first 1.5 years of operations, around 90 fb$^{-1}$ of data were collected. This dataset is used to measure the lifetimes of a few charm hadrons, confirming the expected performance of the Belle II detector, in particular the vertexing, which plays a crucial role in time dependent measurements. Thanks to the performance of the detector and the amount of data that we expect to collect, Belle II will play a crucial role in measuring $CP$ violation and $D^0-\bar{D}^0$ mixing in many decay channels, especially those having neutral particles in the final state. In this presentation we will also show the sensitivity on mixing and CPV parameters in the golden channel $D^0 \to K_S \pi^+ \pi^-$ with a time-dependent Dalitz analysis, and other channels.