You beat me to it. The answer from my science teacher was:
"It will go back in time"
I always thought that was cool. A way to time travel. Made me draw a few spaceships in my tablet.
Excellent question. The answer depends upon what frame of reference you try to measure the projectile's speed. Also, since SR assumes that no craft can travel at the speed of light, we need to assume that the craft is doing something like 99.9% of the speed of light.
From the standpoint of an observer on the craft, the projectile seems to leave the gun in the normal manner, at the normal speed relative to the gun and craft. But to an observer on a planet adjacent to the craft's trajectory, the bullet appears to be moving only a tiny bit faster than the craft, and always less than the speed of light.
This anomaly is better understood if the craft and the ground based observers both have clocks that are observable. To the observer in the craft, he thinks his clock is running at normal speed, but relative to the ground-based observer's clock, the clock on the craft appears to be running v-e-r-y s-l-o-w-l-y. This is called "time dilation," and is an effect predicted by SR, and which has been experimentally verified by comparing atomic clicks on the ground with atomic clocks carried in jet aircraft.
The point of all this is that unlike the Galilean view of the world, in which time was thought to be absolute (all clocks should be in agreement regardless of their frame of reference), the way the Universe actually works is that each observer will measure time in a manner that is dependent upon what his frame of reference is relative to the events he is timing. Essentially, time was thought to be invariant in the Galilean Universe. But as Einstein predicted, and experiment has verified, time is "relative" whereas the invariant quantity is the speed of light: in all frames of reference, light will travel through a vacuum at the same speed.
Use a beam of light instead of a projectile. I asume you'll aim it in the direction that you're already moving. The light will travel at the speed of light, no faster. If you're already traveling at lightspeed when you switch on the beam (which is highly unlikely), the beam will stay with you.