  We prove that a proper weak solution $\{ \Omega_{t} \}_{0 \leq t < \infty}$
to inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n \leq 7$, is smooth
and star-shaped by the time
  \begin{equation*}
  T= (n-1) \log \left( \frac{\text{sinh} \left( r_{+} \right)}{ \text{sinh}
\left( r_{-} \right)} \right), \end{equation*} where $r_{+}$ and $r_{-}$ are
the geodesic out-radius and in-radius of the initial domain $\Omega_{0}$. The
argument is inspired by the Alexandrov reflection method for extrinsic
curvature flows in $\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of
Li-Wei. In addition to this, our methods establish expanding spheres as the
only proper weak IMCF on $\mathbb{H}^{n} \setminus \{ 0 \}$ in all dimensions.
  As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang
and De Lima-Girao to outer-minimizing domains $\Omega_{0} \subset
\mathbb{H}^{n}$ in dimensions $3 \leq n \leq 7$. From this, we also extend a
Penrose-type inequality to balanced asymptotically hyperbolic graphs over the
exteriors of outer-minimizing domains of $\mathbb{H}^{n}$ in these dimensions.
