The geometric structure, electronic structure, magnetic properties and absorption spectrum of graphene-like ZnO (g-ZnO) monolayer supercell with defects are systemically studied by the first-principles calculation based on density functional theory in this work. The defect supercell model includes zinc atom vacancy (V_{Zn_}g-ZnO), oxygen atom vacancy (V_{O_}g-ZnO), nitrogen atom substituted for oxygen atom (N_{O_}g-ZnO) and nitrogen adsorbed on the g-ZnO monolayer (N@g-ZnO). The results indicate that the geometric deformation induced by N-doping in N_{O_}g-ZnO and N@g-ZnO structure is negligible, while that of supercell with vacancy is relatively large. The O atoms neighboring a Zn vacancy center in V_{Zn_}g-ZnO model move away from each other as a result of symmetry breaking. As a contrast, three N atoms around V_O center move into V_{Zn_}g-ZnO supercell. The pristine g-ZnO is non-magnetic. But the magnetic moment of V_Zn_g-ZnO is 2.00 μ_B in total as a result of symmetry breaking. The partial magnetic moment mainly results from the p-orbitals of the three neighboring O atoms. V_{O_}g-ZnO has no magnetic moment, but possesses the electronic structure with identical spin-up and spin-down. The total magnetic moment of the N-doped N_{O_}g-ZnO is 1.00 μ_B, and the total magnetic moment of N@g-ZnO is 3.00 μ_B. Their local magnetic moments are mainly contributed by the p-orbitals of N atom. The density of states and the spin density are given to analyze the magnetic properties. Based on the supercell local symmetry and molecular orbital theory, the origin of magnetic moment is well explained. The magnetic V_{Zn_}g-ZnO, N_{O_}g-ZnO and N@g-ZnO supercell are found to have a D_3h, D_3h and C_3v local symmetry, respectively, which well explains that their total magnetic moments are 2.00 μ_B, 1.00 μ_B and 3.00 μ_B, respectively. The optical absorption characteristics are also discussed. An enhancement of light absorption can be observed for the defective supercells, due to the introduction of defect states into the band gap. The optical transition between gap state and valance band leads to the below band gap absorption. These results are of insightful guidance for understanding properties of graphene-like ZnO monolayer as well as g-ZnO with vacancy and N dopant, and can be theoretically adopted for investigating the nano-electronic devices and catalytic applications based on g-ZnO monolayer.