  The electronic states near the Fermi level of recently discovered
superconductor Ba$_2$CuO$_{4-\delta}$ consist primarily of the Cu $d_{x^2-y^2}$
and $d_{3z^2-r^2}$ orbitals. We investigate the electronic correlation effect
and the orbital polarization of an effective two-orbital Hubbard model
mimicking the low-energy physics of Ba$_2$CuO$_{4-\delta}$ in the hole-rich
regime by utilizing the dynamical mean-field theory with the Lanczos method as
the impurity solver. We find that the hole-overdoped Ba$_2$CuO$_{4-\delta}$
with $3d^8$ (Cu$^{3+}$) is in the orbital-selective Mott phase (OSMP) at
half-filling, and the typical two-orbital feature remains in
Ba$_2$CuO$_{4-\delta}$ when the electron filling approaches $n_e\sim 2.5$,
which closely approximates to the experimental hole doping for the emergence of
the high-$T_c$ superconductivity. We also obtain that the orbital polarization
is very stable in the OSMP, and the multiorbital correlation can drive orbital
polarization transitions. These results indicate that in hole-overdoped
Ba$_2$CuO$_{4-\delta}$ the OSMP physics and orbital polarization, local
magnetic moment, and spin or orbital fluctuations still exist. We propose that
our present results are also applicable to Sr$_2$CuO$_{4-\delta}$ and other
two-orbital cuprates, demanding an unconventional multiorbital superconducting
scenario in hole-overdoped high-$T_c$ cuprates.
