How does the verification process associated with this signature scheme work?
Sorry, can't comment yet. So I'm going be more verbose since it's an answer ;)

I get other values for $z$ and $s$:
$$h^{-1} = 2441\mod q-1 $$ and
$$z = x \cdot h^{-1} = 3612\mod q-1\\
\Rightarrow s = a^z = 1039\mod q$$

The hash value $h$ can be reduced since we
are working modulo $q$. With $q$ prime we have $\phi(q) = q-1$ and
we can reduce the exponent modulo $q-1$. 

This gives
$$y' = s^h = s^{h\mod q-1} \mod q\\$$
So with $h \equiv 2297 \mod q-1$ we verify:
$$y' = 1039^{2297}\mod 6043 = 485 = y$$ 
as desired.