For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions is becoming common practice. In this work we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional model, describing the global characteristics of the circulatory system, and a one-dimensional model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODE's and the 1D hyperbolic system of PDE's), then, we use fixed-point techniques. The abstract result is then applied to the original blood flow case under very realistic hypotheses on the data. This work represents the 1D-0D counterpart of the 3D-0D mathematical analysis reported in [SIAM J. on Multiscale Model. Simul., 1(2) (2003) 173].