Answer
Yes.
Work Step by Step
The bug and the projectile are in an endless free-falling process. So the bug feels no forces exerted on it and its projectile and it floats inside the projectile as if it was in outer space with no gravitational force at all.
The gravitational force here represents the radial force that allows the bug and its projectile ship to continue moving in a circular path around the airless planet.
But, yes, the bug feels weightless.
