The spatial public goods game is one of the most popular models for studying the emergence and maintenance of cooperation among selfish individuals. A public goods game with costly punishment and self-questioning updating mechanism is studied in this paper. The theoretical analysis and Monte Carlo simulation are involved to analyze this model. This game model can be transformed into Ising model with an external field by theoretical analysis. When the costly punishment exists, the effective Hamiltonian includes the nearest-, the next-nearest-and the third-nearest-neighbor interactions and non-zero external field. The interactions are only determined by costly punishment. The sign of the interaction is always greater than zero, so it has the properties of ferromagnetic Ising. The external field is determined by the factor r of the public goods game, the fine F on each defector within the group, and the relevant punishment cost C. The Monte Carlo simulation results are consistent with the theoretical analysis results. In addition, the phase transitions and critical behaviors of the public goods game are also studied using the finite size scaling theory. The results show that the discontinuous phase transition has the same finite size effects as the two-dimensional Ising model, but the continuous phase transitions is inconsistent with Ising model.