Dirac Delta Impulse Response of series RC circuit (Voltage across the capacitor)
The instantaneous rise on the capacitor is not possible in the real world, but then again neither is the impulse it takes to cause that.  The rate of voltage rise on the capacitor is proportional to the input voltage.  During the impulse, that input voltage is infinite, so the capacitor voltage can rise infinitely fast.

This is sortof a word problem illustrating the mathematical concept of limit.  As the input voltage goes towards infinity the capacitor voltage rise time goes towards 0.  In the limit this works out to a finite voltage left on the capacitor, which got there instantaneously.