My reasoning is as follows:
$\mathbb{Z}[X]\Big/(9,18X-3) \cong \left(\mathbb{Z}\Big/\mathbb{9Z}\right)[X]\Big/(\overline{18}X-\overline{3}) = \left(\mathbb{Z}\Big/\mathbb{9Z}\right)[X]\Big/(\overline{6}) \cong \left(\mathbb{Z}\Big/\mathbb{3Z}\right)[X]$
Since $\left(\mathbb{Z}\Big/\mathbb{3Z}\right)[X]$ is an integral domain $(9,18X-3)$ is a prime ideal in $\mathbb{Z}[X]$
Is this correct?