If you have a container that can contain space itself, and a pump that can impel space itself, and you used this combination to pump space out of a container, that container should have more mass than it did when it was full.
So, if you could pump space into the container, then that should mean that it has less mass. Pump enough in, and you may end up with a net negative.
How to contain and pump space is a trivial problem left to the reader.
You can’t randomly swap terms around fields of physics without a complete unifying theory. Here is a link to the paper, because the press is full of click-bait titles, and to show that the author was working in the domain of GR.
As Chronos pointed out, testing existing theories is useful, and even null results help narrow the search for new physics.
In this case this is not a complete model and the author quite clearly states it has issues.
Vectors like used in Newtonian mechanics are OK in spacetime are rotational operations but they are not invariant for boosts. Tensors and Lorentz transformations are just be a hard requirement here if you want to follow this papers math.
This paper is highly speculative and not complete, but that is not an issue and is a sign science is working to still answer questions.
If you want to stick with Newtonian model there is no “negative mass” from that perspective, or if there is weak equivalence principle falls and negative mass proves that inertial and gravitational mass are not equal.
The problem isn’t with this paper or how physicists use abstract concepts or have untested theories. The problem is with people taking their Newtonian assumptions and trying to fit new ideas on to those superseded theories. If these newer ideas perfectly ported back to previous models we simply wouldn’t have replaced those older models.
The entire field knows that GR and QFT are incomplete, but they are both the most accurate theories we have tested to date. It may be that they are incomplete because of the scale. But the lack of a theory of quantum gravity in no way changes the fact that Newtonian mechanics and specifically Galilean transformations are not sufficient to describe this problem.
Here is the pop-science media reporting on this story.
Now compare that to the closing remarks on the above linked paper.
J. S. Farnes should continue to work on his ideas but right now it can’t even approach being an accepted theory as he hasn’t been able to match most of the simple criteria or existing observations, and he doesn’t make that claim.
The press is selling, and the news consumers assuming that there is a lot more to the authors claims and predictions are the problem here. There is no boom and bust here, it is just the media bandwagon that has obscuring the meaning, probably because incremental movement in our understanding doesn’t generate many views.
There are some issues I would ask for clarification from the writer, but nothing that would involve the falsification of the weak equivalence principle like would be suggested if we could map this to Newtonian physics.
It is simply a very speculative theory that is helping meet a critical need in science, testing our assumptions and seeing what we may have missed.
If he can develop it to the point where accurate predictions can be made like the concepts of virtual gravitation or other virtual particles offer and improves on our current models it will be accepted. But at this stage that possibility isn’t even within sight.
Trying to force 
this idea onto F=ma while ignoring very strong tests of the equivalence of gravitational and inertial mass will not get you closer to a more accurate model.
TLDR: Virtual particles are not particles, they are simply a mathematical device that simplifies the formalism. “Negative Mass” is not just a mathematical device and would have broader implications and thus invoking it is held to a higher standard.
A good example of a familiar inertial force is the Coriolis force, which explanes why hurricanes don’t form at the equator.
Assuming that the only difference between a -1kg sphere and a 1 kg sphere is how they bend spacetime, what would happen to the -1kg sphere you dropped standing on Earth?
Careful where you wave that snark.
I think I agree with what you’re trying to say, but words like this are too often taken to mean “Scientists once thought X, they were 100% wrong, now they believe Y”. Or even “So we’re happy teaching kids newtonian physics, even though we know it’s wrong, why do we get so high and mighty about creation science?”
Newtonian physics is used to make millions of accurate predictions every day. It’s still a damn useful model just like the planetary model of the atom, say. The domain of problems that we can use NP to solve is not absolute and it’s fine to point that out in this thread, but there’s a difference between that and implying that NP is wrong or that we cannot still make useful inferences from it.