How to sketch this multirate system spectrum?
You are on the right track.


  After upsampling the spectrum of X would look the same except the x axis would be divided by 2. So the limit of the spectrum would be at pi/4. 


Yes, that's correct.


  Since the filter has amplitude 2 would the amplitude of X be increased from 1 to 2 or will the amplitude of X remain unchanged?


Recall that in the frequency domain, the application of a linear system transfer function $H_B(\Omega)$ is done via straight multiplication of the input signal's spectrum. So yes, since the filter has a magnitude of 2, the amplitude of its output is doubled.


  Finally, for the multiplication of the cosine it would be equivalent to multiply with 2 deltas located at -pi/2 and pi/2. Would the multiplication with this deltas cause a kind of mirroring effect on the spectrum?


Here's where I will make a small correction: multiplication by a cosine in the time domain yields convolution with a pair of Dirac impulse functionss in the frequency domain. Convolution with a Dirac impulse in the frequency domain is equivalent to a shift along the frequency axis. So, you end up with 2 copies of the filter's output spectrum, centered at $\omega = \pm \frac{\pi}{2}$.