Reflection operator for spin 1/2 particle

How do I construct a matrix rep. of a reflection operator for spin 1/2 particle.

Reflection operator ($$\hat R_z$$ : reflection about x-y plane i.e, along z-direction) acts in following way:

$$U^{\dagger}S_z U\rightarrow -S_z,\\ U^{\dagger}S_x U\rightarrow S_x,\\ U^{\dagger}S_y U\rightarrow S_y.$$

I started with assumption: $$U|\uparrow\rangle=c_1|\downarrow\rangle, U|\downarrow\rangle=c_2|\uparrow\rangle$$, and tried to find $$c_1,c_2$$ but no solution satisfy all three constraints!

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.