Consider the flow of an incompressible fluid between two parallel plates, initially induced by a constant pressure gradient. After steady state is attained, the pressure gradient is suddenly with drawn while the plates are impulsively started simultaneously. The arising flow is referred to as run up flow and the present paper aims at studying this flow in the context of a couple stress fluid. Using Laplace transform technique, the expression for velocity is obtained in Laplace transform domain which is later inverted to the space time domain using a numerical approach. The variation of velocity with respect to various flow parameters is presented through graphs.