Stochastic models of chemical reaction networks are an important tool to describe and analyze noise effects in cell biology. When chemical species and reaction rates in a reaction system have different orders of magnitude, the associated stochastic system is often modeled in a multiscale regime. It is known that multiscale models can be approximated with a reduced system such as mean field dynamics or hybrid systems, but the accuracy of the approximation remains unknown. In this paper, we estimate the probability distribution of low copy species in multiscale stochastic reaction systems under short-time scale. We also establish an error bound for this approximation. Throughout the manuscript, typical mass action systems are mainly handled, but we also show that the main theorem can extended to general kinetics, which generalizes existing results in the literature. Our approach is based on a direct analysis of the Kolmogorov equation, in contrast to classical approaches in the existing literature.