We consider the numerical solution of unsteady Stokes equations in patient-specific arterial-like domains in 3D. A Stokes-like solver is a necessary component in a more sophisticated nonlinear Navier-Stokes method, for which several multilevel domain decomposition methods have been introduced recently. Because of the complex geometry, the construction and the solve of the coarse problem usually take a large percentage of the total compute time. In this paper, we introduce a parameterized one-dimensional Stokes solver defined along the centerline of the artery and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable 3D-to-1D restriction and 1D-to-3D extension operators on fully unstructured meshes, a two-level additive Schwarz preconditioner can be constructed. Some numerical experiments for flows in realistic arteries are presented to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than the existing three-dimensional coarse preconditioners.