Partial Fraction Expansion for Inverse Fourier Transform
You should indeed replace $e^{j\omega}$ by the more general complex variable $z$ because when talking about poles and zeros of a rational function, you must consider the whole complex plane. The fact that $e^{-j\omega}=3$ doesn't have a real solution just means that the corresponding pole does not lie on the unit circle.

After having performed the partial fraction expansion, there is nothing that keeps you from setting $z=e^{j\omega}$.