I believe the answer is yes, although I do not know of a reference.
We will use the original definition of pseudo-rationality due to Lipman and Teissier [Lipman–Teissier 1981, p. 102], and the following characterization:
Lemma [Lipman–Teissier 1981, Remark (a) on p. 102]. Let $(R,\mathfrak{m})$ be an $n$-dimensional normal Cohen–Macaulay local ring. Then, $R$ is pseudo-rational if and only if for every projective birational morphism $f\colon W \to \operatorname{Spec}(R)$ where $W$ is integral, there exists a proper birational morphism $g\colon W' \to W$ such that $W'$ is normal and integral and such that
$$\delta_{fg}\colon H^n_{\mathfrak{m}}(R) \longrightarrow H^n_{(fg)^{-1}(\{\mathfrak{m}\})}(\mathcal{O}_{W'})$$
is injective.
Proposition. Let $(R,\mathfrak{m})$ be an $n$-dimensional noetherian local ring that is pseudo-rational, and let $\varphi\colon R \to S$ be an étale map. Then, for every prime ideal $\mathfrak{n} \subseteq S$ lying over $\mathfrak{m}$, the ring $S_{\mathfrak{n}}$ is pseudo-rational.
Proof. Since $\varphi$ is étale, $S$ is normal and Cohen–Macaulay [Matsumura 1989, Corollary to Theorem 23.9 and Corollary to Theorem 23.3], and $S_\mathfrak{n}$ is of dimension $n$ [Stacks, Tag 04N4].
Now let $f_\mathfrak{n}\colon W_\mathfrak{n} \to \operatorname{Spec}(S_\mathfrak{n})$ be a projective birational morphism, where $W_\mathfrak{n}$ is integral. Since $S_{\mathfrak{n}}$ is integral and noetherian, there exists an ideal $I_{\mathfrak{n}} \subseteq S_{\mathfrak{n}}$ such that $W_\mathfrak{n}$ is the blowup along $I_{\mathfrak{n}}$ [EGAIII$_1$, Corollaire 2.3.6]. Clearing denominators, there exists an ideal $I \subseteq S$ localizing to $I_{\mathfrak{n}}$ such that the blowup $f\colon W \to \operatorname{Spec}(S)$ along $I$ localizes to $f_\mathfrak{n}$.
By [Stacks, Tag 087B] or [Rydh, Proposition 4.14], there exists an ideal $J \subseteq R$ such that we have the commutative diagram
$$\require{AMScd}\begin{CD}
W' @>g>> W @>f>> \operatorname{Spec}(S)\\
@VVV @. @VVV\\
\operatorname{Bl}_J\bigl(\operatorname{Spec}(R)\bigr) @= \operatorname{Bl}_J\bigl(\operatorname{Spec}(R)\bigr) @>h>> \operatorname{Spec}(R)
\end{CD}$$
where $W' \to \operatorname{Spec}(S)$ is the blowup along $JS$, and the outer rectangle is cartesian by flat base change [Stacks, Tag 0805]. Replacing $\operatorname{Bl}_J(\operatorname{Spec}(R))$ by its normalization, we may assume that $\operatorname{Bl}_J(\operatorname{Spec}(R))$ is normal.
Now by flat base change for local cohomology [Hashimoto–Ohtani 2008, Theorem 6.10], the homomorphism
$$\delta_{f_\mathfrak{n}g_\mathfrak{n}}\colon H^n_{\mathfrak{n}}(S_{\mathfrak{n}}) \longrightarrow H^n_{(f_\mathfrak{n}g_\mathfrak{n})^{-1}(\{\mathfrak{n}\})}(\mathcal{O}_{W'_{\mathfrak{n}}})$$
is obtained from $\delta_h$ by tensoring with $S_{\mathfrak{n}}$, and hence is injective. Finally, since $fg$ is a blowup and $W'$ is normal by [Matsumura 1989, Corollary to Theorem 23.9], we see that $S_\mathfrak{n}$ is pseudo-rational by the Lemma above. $\blacksquare$
It should be possible to prove that completions of pseudo-rational G-rings are pseudo-rational, using Néron–Popescu desingularization [Popescu 1986, Theorem 2.4; Popescu 1990, p. 45; Swan 1998, Theorem 1.1] to reduce to showing that pseudo-rationality is preserved under étale extensions (shown above) and polynomial extensions.
