Impulse Response from Frequency Response at Specific Values
If you look at the definition of $H(j\omega)$

$$H(j\omega)=\sum_{n=0}^{N-1}h(n)e^{-jn\omega}$$

then you can see that

$$H(0)=\sum_{n=0}^{N-1}h(n)=h(0)+h(1)+h(2)+h(3)$$

and

$$H(j\pi/2)=\sum_{n=0}^{N-1}h(n)e^{-jn\pi/2}=h(0)-jh(1)-h(2)+jh(3)$$

and

$$H(j\pi)=\sum_{n=0}^{N-1}h(n)e^{-jn\pi}=h(0)-h(1)+h(2)-h(3)$$

This gives you 4 equations for the 4 unknown coefficents:
$$h(0)+h(1)+h(2)+h(3)=2\\
h(0)-h(2)=7\\
-h(1)+h(3)=-3\\
h(0)-h(1)+h(2)-h(3)=0$$

I guess you can take it from here.