Careful, spin is a pseudo vector. Therefore the the reflection operator rather acts like ($P$ like parity, $R$ usually represents rotation operators):
$$
\begin{align}
P_z&: & S_x&\to-S_x & S_y&\to-S_y & S_z&\to S_z
\end{align}
$$
This can be implemented with a $\pi$ rotation about the $z$ axis, ie:
$$
U=e^{i\pi S_z}=2S_z
$$
In general, the spin $1/2$ representation is entirely determined by the spin operators that must satisfy the commutation relation ($\hbar=1$):
$$
[S_i,S_j]=i\epsilon_{ijk}S_k
$$
In fact, an automorphism can always be uniquely represented (up to a phase) by a rotation.
Note that in your case, the commutations switch signs, so such a $U$ is impossible to find. You were actually looking for the representation of reflection followed by time reversal $P_z T$. $T$ is defined by switching the sign of spin:
$$
\begin{align}
T&: & S_x&\to-S_x & S_y&\to-S_y & S_z&\to -S_z
\end{align}
$$
The latter has to be represented as an antiunitary transformation (conserves inner product, but antilinear). This explains why the commutation relations switch signs as well.
Hope this helps.