Photoneutron emission cross sections were measured for $^{13}$C below $2n$ threshold using quasi-monochromatic $\gamma$-ray beams produced in laser Compton-scattering at the NewSUBARU synchrotron radiation facility. The data show fine structures in the low-energy tail of the giant-dipole resonance; the integrated strength of the fine structure below 18~MeV is intermediate among the past measurements with bremsstrahlung and the positron annihilation $\gamma$ rays. We compare the photoneutron emission data with the {\sf TALYS} statistical model calculation implemented with the simple modified Lorentzian model of $E1$ and $M1$ strengths. We also compare the total photoabsorption cross sections for $^{13}$C with the shell model and antisymmetrized molecular dynamics calculations as well as the statistical model calculation. We further investigate the consistency between the present photoneutron emission and the reverse $^{12}$C(n,$\gamma$) cross sections through their corresponding astrophysical rate.
