  We investigate baryogenesis via leptogenesis in $A_4$ flavor model within the
paradigm of type-I and II seesaw mechanism resulting in magic neutrino mass
matrix with broken $\mu-\tau$ symmetry in a minimal scenario with two
right-handed neutrinos(2RHN). Additional $Z_3$ cyclic symmetry is employed to
constrain the Yukawa structure of model. The type-II seesaw terms play crucial
role in generating non-degenerate neutrino masses and non-zero $\theta_{13}$
and contribute in baryogenesis. In particular, after the spontaneous symmetry
breaking, the Yukawa couplings $y_{\Delta_1}$ and $y_{\Delta_3}$ are
responsible for the breaking of $\mu-\tau$ symmetry. The effective Majorana
neutrino mass $|M_{ee}|$ is found to be well within the sensitivity reach of
the $0\nu\beta\beta$ experiments, in particular, for inverted hierarchy. The
model has imperative implication for inverted hierarchy, for example, the
non-observation of this process at nEXO will rule out IH. The predicted baryon
asymmetry is in good agreement with the observed baryon asymmetry for NH
whereas IH is disallowed at 2.5$\sigma$ C.L..
