We consider a Navier-Stokes equation in two and three space dimensions subject to periodic boundary conditions and perturbed by a transport type noise. The perturbation is sufficiently smooth in space, but rough in time. The system is studied within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of weak solution to the Navier-Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we also present uniqueness and stability results with respect to the driving signal.