  We study the effects of random nonmagnetic impurities on superconducting
transition temperature $T_c$ in a Cu-doped Bi$_2$Se$_3$, for which four types
of pair potentials have been proposed. Although all the candidates belong to
$s$-wave symmetry, two orbital degree of freedom in electronic structures
enriches the symmetry variety of a Cooper pair such as even-orbital-parity and
odd-orbital-parity. We consider realistic electronic structures of Cu-doped
Bi$_2$Se$_3$ by using tight-binding Hamiltonian on a hexagonal lattice and
consider effects of impurity scatterings through the self-energy of the Green's
function within the Born approximation. We find that even-orbital-parity
spin-singlet superconductivity is basically robust even in the presence of
impurities. The degree of the robustness depends on the electronic structures
in the normal state and on the pairing symmetry in orbital space. On the other
hand, two odd-orbital-parity spin-triplet order parameters are always fragile
in the presence of potential disorder.
