  Consider the Ising model at low-temperatures and positive external field
$\lambda$ on an $N\times N$ box with Dobrushin boundary conditions that are
plus on the north, east, and west boundaries and minus on the south boundary.
If $\lambda = 0$, the interface separating the plus and minus phases is
diffusive, having $O(\sqrt N)$ height fluctuations, and the model is fully
wetted. Under an order one field, the interface fluctuations are $O(1)$ and the
interface is only partially wetted, being pinned to its southern boundary. We
study the critical pre-wetting regime of $\lambda_N \downarrow 0$, where the
height fluctuations are expected to scale as $\lambda^{ -1/3}$ and the rescaled
interface is predicted to converge to the Ferrari--Spohn diffusion. Velenik
(2004) identified the order of the area under the interface up to logarithmic
corrections. Since then, more refined features of such interfaces have only
been identified in simpler models of random walks under area tilts.
  In this paper, we resolve several conjectures of Velenik regarding the
refined features of the Ising interface in the critical pre-wetting regime. Our
main result is a sharp bound on the one-point height fluctuation, proving $e^{
- \Theta(x^{3/2})}$ upper tails reminiscent of the Tracy--Widom distribution,
capturing a tradeoff between the locally Brownian oscillations and the global
field effect. We further prove a concentration estimate for the number of
points above which the interface attains a large height. These are used to
deduce various geometric properties of the interface, including the order and
tails of the area it confines, and the poly-logarithmic pre-factor governing
its maximum height fluctuation. Our arguments combine classical inputs from the
random-line representation of the Ising interface, with novel local resampling
and coupling schemes.
