Negative Mass is Very Confusing
I'll just start with this: the behavior of negative mass would be really, really confusing, assuming many of the models we use in physics apply here. (Which is what I'm doing.)
Mathematically, it's easy to compensate for: you just throw a negative sign wherever you see mass. Understanding how that negative sign affects things, however, is what really boggles the mind.
For instance, if you push it, it pushes back, but not only in the Newton's Laws sort of way. You throw a (normal) ball at a lump of this stuff, and the matter pushed into the ball instead of moving with it. Because of its negative mass, the force you apply goes negative: it experiences a pull when normal matter experiences a push.
It falls up.
You push from behind to make it slow down.
Friction can make it speed up. (Surface friction, anyways, because it's based on the normal force, which is based off of the object's mass, which is now negative!)
It still follows the right-hand rule of electromagnetism. Okay, that isn't abnormal on its own, but if certainly feels abnormal given the other odd things this matter does. This indifference to electrical forces to will be the key to manipulating and shooting this stuff.
Negative Mass Projectiles
If a negative-mass thing has electric charge (or can be induced to have electric charge), you can use something like the Lorentz-Force and make a railgun-like weapon. Normal accelerants, like gunpowder, would cause the negative-mass projectile to fly into the center of the explosion! I'm not sure how this would work with a barrel, where this would compress the exploding gases further, but I can say with reasonable certainty this is a bad idea.
What happens when it collies with something? The (positive) mass object reacts as normal. The negative mass projectile will, instead of bouncing off, stay with the object. Indeed, running into an object may even increase it's speed! This is because the reaction force, which normally causes a projectile to slow, will actually cause it to speed up instead!
Of course, this is speculation: we don't have negative mass objects around to do tests on. Just like many other things, we will not know for sure until we test it. Maybe negative mass objects have different physics that the standard models doesn't account for?
Negative Energy?
"Negative energy" doesn't really mean much. It just means the energy involved is going out, or is lower than what you think "0" ought to be. For instance, if I measure potential energy from the top of Mt. Everest, we all have negative potential energy. (Unless you are reading this on top of Everest or higher... in which case, I guess you do have positive potential energy.) Energy levels of electrons in atoms are usually assigned negative energy levels. "Negative Energy" weapons won't do anything unusual compared to your negative mass projectile.
