Studying out-of-equilibrium physics in quantum systems under quantum quench is of vast experimental and theoretical interests. Using periodic quantum quenches, we present an experimentally accessible scheme to simulate the quantum Zeno and anti-Zeno effects in an open quantum system of a single superconducting qubit interacting with an array of transmission line resonators. The scheme is based on the following two observations: Firstly, compared with conventional systems, the short-time non-exponential decay in our superconducting circuit system is readily observed; and secondly, a quench-off process mimics an ideal projective measurement when its time duration is sufficiently long. Our results show the active role of quantum quench in quantum simulation and control.