We are given $\overline z + i\overline w = 0.$

So, $\overline z = -i\overline w.$

Take conjugate on both sides: $z = \overline{-i\overline w} = i w,$ because $\overline{-i}=i$ and $\overline{\overline w}=w$.

Hence, $z = iw.$

Now write arguments: $\arg z = \arg(iw) = \arg i + \arg w = \frac{\pi}{2} + \arg w \pmod{2\pi}.$

Also given: $\arg(zw)=\pi.$

Since $z=iw$, $zw = (iw)w = iw^2.$

Therefore, $\arg(zw)=\arg i + 2\arg w = \frac{\pi}{2} + 2\arg w = \pi \pmod{2\pi}.$

Solve for $\arg w$: $\frac{\pi}{2}+2\arg w=\pi$ $2\arg w=\frac{\pi}{2}$ $\arg w=\frac{\pi}{4} \pmod{\pi}.$

Taking the principal value consistent with the options, $\arg w=\frac{\pi}{4}.$

Then $\arg z=\frac{\pi}{2}+\frac{\pi}{4}=\frac{3\pi}{4}.$

Check options:

• A: $\frac{5\pi}{4}$
• B: $\frac{\pi}{2}$
• C: $\frac{3\pi}{4}$
• D: $\frac{\pi}{4}$

So the correct option is $\boxed{\frac{3\pi}{4}}.$