Powering buck converter using charged capacitor
Yea, of course it is.
In a buck converter, the switch (you have drawn) opens and closes at a certain rate and duty cycle. Current is only consumed from the input (capacitor) during the on-time of this switch. What happens during the off-time is irrelevant, and you could use that to recharge the capacitor.
This is not necessarily a good idea; you need additional switches, and in most circumstances, it doesn't perform any better than just leaving the input voltage source permanently attached.