  Two-dimensional ($2$D) semi-Dirac systems, such as $2$D black phosphorus and
arsenene, can exhibit a rich topological phase transition between insulating,
semi-Dirac, and band inversion phases when subjected to an external modulation.
How these phase transitions manifest within the quantum transport and shot
noise signatures remain an open question thus far. Here, we show that the Fano
factor converges to the universal $F\approx0.179$ at the semi-Dirac phase, and
transits between the sub-Poissonian ($F\approx1/3$) and the Poissonian shot
noise ($F\approx1$) limit at the band inversion and the insulating phase,
respectively. Furthermore, the conductance of $2$D semi-Dirac system converges
to the contrasting limit of $G/G_0 \rightarrow 1/d$ and $G/G_0 \rightarrow0$ at
the band inversion and the insulating phases, respectively. The quantum
tunneling spectra exhibits a peculiar coexistence of massless and massive Dirac
quasiparticles in the band inversion regime, thus providing a versatile sandbox
to study the tunneling behavior of various Dirac quasiparticles. These findings
reveal the rich interplay between band topology and quantum transport
signatures, which may serve as smoking gun signatures for the experimental
studies of semi-Dirac systems near topological phase transition.
