Within the effective Lagrangian approach, we study the $p\bar{p} \rightarrow \psi\eta$ [$\psi\equiv\psi(3686)$, $J/\psi$] reaction at the low energy where the contributions from nucleon pole and low-lying nucleon resonances, $N(1520)$, $N(1535)$ and $N(1650)$ are considered. All the model parameters are determined with the help of current experimental data on the decay of $\psi \to p \bar{p} \eta$... Within the model parameters, the total and differential cross sections of the $p\bar{p} \rightarrow \psi\eta$ reaction are predicted. We show that the relative phases between different amplitudes of different nucleon resonance will change significantly the angular distributions of the $\bar{p} p \to \psi \eta$ reaction. Therefore, we conclude that these reactions are suitable to study experimentally the properties of the low-lying nucleon resonance and the reaction mechanisms. We hope that these theoretical calculations can be tested by future experiments. (read more)