The charge exchange spin-dipole (SD) excitations of $^{90}$Zr are studied in the Skyrme Hartee-Fock plus proton-neutron random phase approximation with SAMi-J interactions. Experimental value of the model-independent sum rule obtained from the SD strength distributions of $^{90}$Zr(p,n)$^{90}$Nb and $^{90}$Zr(n,p)$^{90}$Y is used to deduce the neutron skin thickness. The neutron skin thickness $\Delta r_{np}$ of $^{90}$Zr is extracted as $0.083\pm0.032$ fm, which is similar to the results of other studies. Based on the correlation analysis of the neutron skin thickness $\Delta r_{np}$ and the nuclear symmetry energy $J$ as well as its slope parameter $L$, a constraint from the extracted $\Delta r_{np}$ leads to the limitation of $J$ to $29.2 \pm 2.6$ MeV and $L$ to $53.3 \pm 28.2$ MeV.