  We have carried out exclusive measurements for the photoproduction of an
$\eta$ meson from a proton target with an egg-shaped calorimeter made of BGO
crystals (BGOegg) and forward charged-particle detectors at the SPring-8 LEPS2
beamline. The differential cross sections and photon beam asymmetries of the
$\gamma p \to \eta p$ reaction are measured in a center-of-mass energy ($W$)
range of $1.82$-$2.32$ GeV and a polar angle range of $-1.0 <
\cos{\theta^{\eta}_{\mathrm{c.m.}}} < 0.6$. The reaction is identified by
selecting a proton and two $\gamma$'s produced by an $\eta$-meson decay. The
kinematic fit method was employed to select the reaction candidate with the
confidence level larger than $1$\%. A bump structure at $W$ = $2.0$-$2.3$ GeV
in the differential cross section is confirmed at extremely backward $\eta$
polar angles, where the existing data are inconsistent with each other. This
bump structure is likely associated with high-spin resonances that couple with
$s\bar{s}$ quarks. The results of the photon beam asymmetries in a wide $\eta$
polar angle range are new for the photon beam energies exceeding $2.1$ GeV.
These results are not reproduced by the existing partial wave analyses. They
provide an additional constraint to nucleon resonance studies at high energies.
