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.
Comment: 16 pages, 11 figures