It is proved (but not explicitly stated) in
Heinrich, Stefan Ultraproducts in Banach space theory. J. Reine Angew. Math. 313 (1980), 72–104.
The result you want follows from what Heinrich calls "the local reflexivity of ultrapowers", which is Theorem 7.3 in the paper.
EDIT: Here is how to apply Heinrich's Theorem 7.3. It is an elementary observation that an $n$ dimensional subspace $E$ of a Banach space $Y$ is $C$-complemented iff there is an $n$ dimensional subspace $F$ of $Y^*$ that $C$-norms $E$. Suppose that $E \subset X^U$ is $n$ dimensional and $F \subset (X^U)^*$ is $n$ dimensional and $C$-norms $E$. If $F\subset (X^*)^U$, there is no problem pulling back to $X$, and Heinrich's local reflexivity principle for ultrapowers says that you can take $F\subset (X^*)^U$ (which is strictly smaller than $(X^U)^*$ when $X$ is not superreflexive).