Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit.

All the polynomials of the form $$1+2p(x)$$ are units. This is because $2p(x)$ is nilpotent, and elements of the form $1+n$, $n$ nilpotent, are units in any ring.

These are all the units of $\mathbf{Z}_4[x]$. This follows from the fact that if $u(x)$ is a unit, then it must remain a unit after being reduced modulo two. But the only unit of $\mathbf{Z}_2[x]$ is the constant $1$.