Casimir Vacuum gravity propulsion? This question is quite similar to Poor man's Alcubierre drive? but definitely not a duplicate since the direction of movement, in my proposal, is the opposite. This is also similar but different. (does not involve gravity) It is an highly speculative subject so maybe it will be not allowed.
Let us consider the Archimedes Experiment, conducted by INFN in Italy: for now the results are described as 'encouraging' so it looks like positive mass repeals Casimir Vacuum. (you could look at this question as an analogous model).
If so: connecting a positive mass to a pair on plates, the whole system will move in the direction from the plates to the mass?
Impulse is conserved because virtual pairs between Casimir plates should be accelerated in the direction opposite to direction of the mass. (as in the analogous fluid dynamics model I linked)
 A: Whenever you have an attractive, conservative force between objects, the potential energy is smaller when the objects are closer together. Potential energy is energy and energy is mass, so the gravitational mass of the system is reduced when the objects are moved closer together. This is true regardless of the details of the force (and the Casimir force is just an ordinary force).
If you adopt the convention that the potential energy is zero at infinite separation, then it's negative at any finite separation. That is no way implies that there's a nonzero energy density in the GR stress-energy tensor in any region of spacetime.
The usual Casimir effect is a force between conducting plates. Real conducting plates have a nonzero EM field that extends past their boundaries, which contributes a positive energy density to the "vacuum" between them. It's been argued that there is still, in spite of that, a locally negative energy density in part of the region between the plates at sufficiently large separations. I distrust all of these calculations (in this one, the fact that they find the negative Casimir energy density is independent of position). I tend to think that the apparent prediction of negative energy densities in some configurations in QFT probably just indicates there's something wrong with the theory or the calculation, but that's just a guess/prejudice.
In the INFN Archimedes experiment, their "plates" seem to be separated by atomic distances in a crystal. I'm pretty sure that the energy density is demonstrably everywhere (very) positive in this setup, and all they're showing is that there is no violation of energy conservation or the equivalence principle. Even calling this the Casimir force seems dubious, and I wonder if it's just a buzzword to get funding.

That aside, you can still ask what would happen in principle if you were able to construct an object with a negative mass, and put it next to an object with positive mass. I'll treat the problem in Newtonian gravity, since I can't treat it exactly in GR and I don't expect that the behavior predicted by GR would be much different.
The gravitational forces on the objects are equal and opposite as always, and outward directed since the product of their masses is negative. But the acceleration is opposite the force for the negative-mass object, so both objects accelerate in the same direction.
There is no violation of momentum conservation since $\sum F=d(\sum p)/dt=0$. If the sum of the masses is zero, there's no violation of energy conservation either since the total kinetic energy is zero.
If the sum of the masses is nonzero, the objects will have different accelerations and the distance between them will change. You can presumably say that the kinetic energy change comes from a potential energy which is a function of the separation of the objects.
You could connect a rigid strut between the unequal masses to try to keep their separation constant. The trouble is that the strut enforces the separation by applying equal and opposite restoring forces to the objects, and the negative-mass object will react to that by accelerating in the opposite direction of the "intended" restoring force. This should counter the net gravitational acceleration so the masses connected by the strut don't accelerate at all.
Despite the weirdness of the setup, the analysis doesn't seem much different from the case when both masses are positive.
