Inverse Discrete-Time Fourier Transform of $X(Ω)=jΩ$
This system is a discrete-time differentiator, and in this case it's easier if you directly solve the inverse DTFT integral:

$$x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(\omega)e^{jn\omega}d\omega=\frac{j}{2\pi}\int_{-\pi}^{\pi}\omega\, e^{jn\omega}d\omega\tag{1}$$

Using integration by parts it's quite straightforward to solve $(1)$. The result should be

$$x[n]=\begin{cases}0,\quad n=0\\\displaystyle\frac{(-1)^n}{n},\quad\text{otherwise}\end{cases}$$