  We present a measurement of the cross section and transverse single-spin
asymmetry ($A_N$) for $\eta$ mesons at large pseudorapidity from
$\sqrt{s}=200$~GeV $p^{\uparrow}+p$ collisions. The measured cross section for
$0.5<p_T<5.0$~GeV/$c$ and $3.0<|\eta|<3.8$ is well described by a
next-to-leading-order perturbative-quantum-chromodynamics calculation. The
asymmetries $A_N$ have been measured as a function of Feynman-$x$ ($x_F$) from
$0.2<|x_{F}|<0.7$, as well as transverse momentum ($p_T$) from
$1.0<p_T<4.5$~GeV/$c$. The asymmetry averaged over positive $x_F$ is
$\langle{A_{N}}\rangle=0.061{\pm}0.014$. The results are consistent with prior
transverse single-spin measurements of forward $\eta$ and $\pi^{0}$ mesons at
various energies in overlapping $x_F$ ranges. Comparison of different particle
species can help to determine the origin of the large observed asymmetries in
$p^{\uparrow}+p$ collisions.
