The interpretation of higher-order logic depends of course
on the set-theoretic background in which it is computed.
And if one is willing to change this set-theoretic
background, then one may arrive at an answer to the
question.

The situation is that whenever one is considering the
ultrapower of a given structure $\cal M$ by an ultrafilter
$U$ on $I$, then one might simultaneously consider the
ultrapower of all other structures $\cal N$ by $U$, and
realize that all such ultrapowers and ultraproducts by $U$
fit together in a coherent way, given by the ultrapower of
the entire set-theoretic universe by $U$. In particular, one may take the ultrapower of the higher-order structures built from $\cal M$ by applying the power set operation, as indicated by Gerhard, and conclude a higher-order analogue of the fact you mention in the question. Indeed, one may apply the power set transfinitely.

Specifically, if you
have an ultrafilter $U$ on a set $I$, then one may form the ultrapower of the entire set-theoretic universe $V$ by building the structure $\bar V=V^I/U$, consisting of the appropriate
equivalence classes $[f]_U$, where $f:I\to V$ is any
function on $I$ to set-theoretic objects $f(i)$. Each point
$[f]_U$ amounts to the ultraproduct $(\Pi_i f(i))/U$, and
in this way each ultraproduct construction is seen as a
special case of the ultrapower of the universe (reversing
the usual description of ultrapowers as special cases of
ultraproducts).

By Los's theorem, the canonical map $j:V\to M$ mapping each
object $x$ to $[x]_U$, which is essentially the ultrapower
of $x$ by $U$, is an elementary embedding in the language
of set theory. This implies, in particular, that $\bar V$
is a model of the axioms of set theory, and furthermore,
that any set-theoretic statement about any structure $\cal
M$, including higher-order statements of any higher order,
including transfinite order, is preserved from the
structure $\cal M$ to the image $j({\cal M})$, as
interpreted in the new set-theoretic universe $\bar V$.
Thus, higher order statements about $\cal M$ in $V$ are
directly preserved to the corresponding higher-order
statements about the ultrapower of $\cal M$, but
interpreted now in $\bar V$.