Same with normal gravity, but backwards.
Both mass would experience negative repulsive force:
F=Gm(-m)/r^2
=-Gm^2/r^2
The positive mass accelerates away from the negative mass, because F=ma
=> ma=-Gm^2/r^2
=> a=-Gm/r^2
Now here's the kicker. If inertial mass (the m in F=ma), is the same thing as gravitational mass. Then, the negative mass accelerates towards the positive mass. Because
F=(-m)a
=> F=-ma
=> -ma=-Gm^2/r^2
=> a=Gm/r^2
So accelerates in same direction as positive mass.
Technically, conservation of energy & momentum is maintained, because the negative kinetic energy of negative mass would cancel out positive energy of positive mass.
But I think the above can't be natural, because the magnitude of kinetic energy would grow to infinity, leading to even more impossibilities.
So you'd need to 'hack' the theory of gravity and make inertial mass different to gravitational mass. So the particle has negative gravitational mass, but positive inertial mass, to make laws of physics stay sane :)
Now, the kinetic energy of the positive mass grows, but the kinetic energy of the negative mass is negative and falls (because its mass is negative...), so energy is conserved.
EDIT to add: I think I gathered wrong. Apparently, there is runaway motion (same direction), energy and momentum is still conserved (since one mass is negative), and since total mass is zero, the two particles reach speed of light. Intriguing. See paper.
"Firstly, the theory of positive–negative mass particle pairs provides clear rules that govern such interactions. The mechanics of these interactions are governed by the usual physical laws: the conservation of energy and momentum remain fundamental, and hence it is unclear why we should object to this potentially physical law of nature on grounds of aversion alone. Secondly, and more importantly, observations provide evidence for significant numbers of ultra-high-energy cosmic rays which are known to be extragalactic in origin, although the mechanism of their production remains a mystery (Pierre Auger Collaboration 2017). From this perspective, runaway motion is not a challenge for negative mass models, but is rather a useful observational constraint. The idea that all negative masses in a universe should form gravitational dipoles and accelerate to high energies is not supported by the simulations presented here (which have a limited number of particles), in which no runaway particles can be identified. While runaway motion is a legitimate physical facet of negative mass particle interactions, the simulations indicate that this behaviour is only common for idealised particle pairs and occurs more rarely as a bulk behaviour within a negative mass fluid. This is likely as the particles in such a fluid are subject to numerous counteracting forces from the surrounding medium. One can assume that some amount of runaway particles must still exist, although these would likely be highly scattered by Brownian motion (e.g. Landis 1991)."