  We study the (001) surface state of a recently proposed $E_g$ symmetry
inter-orbital-odd spin-triplet s-wave superconducting (SC) state in
Sr$_2$RuO$_4$ (SRO). We confirm that this pair potential is transformed into a
chiral $d$-wave pair potential and a pseudo-Zeeman field in the band basis for
a low-energy range. Due to the chiral $d$-wave pair potential, the surface
states appear near zero energy in the momentum range enclosed by the nodal
lines of the chiral d-wave pair potential for each band at the (001) surface.
Nevertheless, the pseudo-Zeeman field gives band splitting of the surface
states, and its splitting energy is much smaller than the SC energy gap. The
local density of states (LDOS) at the (001) surface of the SC state has a
pronounced peak structure at zero energy due to the surface states near zero
energy when the order of the resolution is lower than the splitting energy.
This peak structure is robust under perturbations, such as an orbital Rashba
coupling or an $E_u$ SC pair potential at the surface.
