$$H(\Omega)=\begin{cases}\exp(-j \pi/2) ,\;&\Omega >0 \\ \exp(j\pi/2) ,\;&\Omega<0.\end{cases}$$
How can I find the inverse fourier transform of this $(h(t))$ using fourier properties?
Fourier Transform Used: $$X(\Omega)\equiv\int_{-\infty}^{\infty}x(t)\,e^{-j\Omega t}\,dt.$$
First, to simplify notation a bit, we notice that, since $e^{j\pi/2}=j,$ it follows that $$H(\Omega)=-j\operatorname{sgn}(\Omega). $$ In looking at the wiki table of Fourier Transforms, in the non-unitary, angular frequency column (which corresponds to the definition you're using), we see that the tricky part here is to note that $$\mathcal{F}^{-1}(-j\pi\operatorname{sgn}(\Omega))=\frac1t,$$ so that $$\mathcal{F}^{-1}(-j\operatorname{sgn}(\Omega))=\frac{1}{\pi t},$$ since the Inverse Fourier Transform is linear. However, this does not take into account the situation in which $t=0$. Technically, your $H(\Omega)$ is undefined there, unless you made a typo in your definition and one of the inequalities is non-strict.