Developing metal-free oxygen redox catalysts for oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) to eliminate the risk caused by transition metal based catalysts is highly desirable as promising substitutes for precious metal catalysts for widespread adoption of rechargeable Zn-Air batteries. Herein, N, S co-doped carbon (NS/C) was developed by a molecular-level doping strategy using a new precursor of N and S co-containing bis(imino)-pyridine based polymer, copolymerized by three monomers of 2, 6-Diacetylpyridine (DAP), 1, 5-naphthalenediamine (NDA) and 2, 5-dithiobiurea (DBU) with Schiff base reaction. The results demonstrated that the ratio between the three monomers strongly affects the texture characteristics of the corresponding carbon materials, doping types and contents of N and S, as well their synergetic effects in carbon matrix. The NS/C obtained by the copolymer of DAP, NDA and DBU with their corresponding molar ratio of 1: 0.5: 0.5 exhibits a small overpotential difference of 0.72 V vs RHE and a remarkable stability in 0.1 M KOH for overall ORR/OER activity, outperforming most of metal-free catalysts reported. Meanwhile, a single rechargeable Zn-Air battery test further identified its excellent activity, in which the battery delivers a maximum power density output of 149 mW·cm−2, a specific capacity of 769 mAh·gZn−1, and long charge-discharge capability at 5 mA·cm−2 with 600 cycles in 100 h, surpassing that with the mixed Pt/C and IrO2 (1:1 wt.%) catalysts under the same mass loading. The theoretical calculation results show that the oxygen redox activity of NS/C critically roots in C atoms with positive charge and spin density, induced by the strong interaction between the specific N and S dopants derived from the new designed precursors.