A weakening of superconductivity upon substitution of Cu by Zn (0.5~1 %) is observed in a high-T_c cuprate, Ca_{2-x}Na_xCuO2Cl2, near the hole concentration of 1/8 per Cu. The superconducting transition temperature and its volume fraction, estimated by magnetic susceptibility, exhibit a sizable anomaly for x=0.12~0.14, where the slowing down of Cu spin fluctuations below 5 K is demonstrated by muon spin relaxation experiments. These observations are in close resemblance to other typical cuprates including YBa2Cu3O_{7-d}, and Bi2Sr2Ca_{1-x}Y_xCu2O_{8+d}, providing further evidence that Zn-induced stripe correlation is a universal feature of high-T_c cuprate superconductors common to that of La_{2-x}A_{x}CuO4 (A=Ba, Sr).