Fictional (But Realistic) Spacecraft

[T. A. Gardner]

No it won't. You just failed basic physics.
https://en.wikipedia.org/wiki/Inelastic_collision
If it feels something you would get a meaningful recoil. But everything being equal regardless of collision physics type used the difference is purely the mass ratio between object_1 and object_2.
In this case a 4 grams projectile verses a ship ranging from 0.5 to 100 million metric tons depending on crew size and mission duration: 8e-12 to 4e-15

At this point you're deliberately just throwing baseless random thoughts. I see no point continuing.

What I would be worried about isn't recoil, but penetration. A hole in your spaceship, even a very small one, is a very bad thing... Putting enough armor on the front of your spaceship as it zips through space is easier if you don't have a lot of 'front' to put it on...