We study an abelian $U(1)_{X}$ extension to the Standard Model group consisting of an extended scalar sector of two doublets and one singlet plus three additional exotic quarks and two exotic leptons through non-universal interaction of the new $X$ quantum number and a $\mathbb{Z}_{2}$ parity. In this model, the lightest fermions are massless at tree-level so effective operators up to dimension seven are considered to fill all zeros in mass matrices, providing an upper bound for the $\Lambda$ energy scale from the electron, up, down and strange quark masses. We obtain predictions on the anomalous $B$ meson decay showing that it can explain the anomaly while it provides an upper bound for $v_{\chi}$ of $21$ TeV which translates in an upper bound for the new $Z_{2}^{\mu}$ gauge boson of $25$ TeV. Finally, muon $g-2$ contributions are also studied, where we found that positive contributions coming from charged $W^{+}$ bosons and exotic Majorana neutrinos while contributions coming from heavy charged and neutral scalar bosons, are negative. In particular, the allowed regions according to the experimental muon $g-2$ parameter, shows that as the exotic neutrino mass increases, heavy scalars and the exotic lepton masses increase as well.