If one posutlates a point negative gravitational mass, an effective potential analysis (such as the one at http://www.fourmilab.ch/gravitation/orbits/
shows that it would require infinite energy to reach r=0. One could come as close to the central point mass as desired, but never reach it with a finite "energy at infinity".
But if one postualtes a distributed, rather than a point, mass, this issue goes away. A distributed negative gravitational mass would have a metric that was well-behavied everywhere, and one could reach the center of it without any special problems.
So there wouldn't be any event horizon, nor would there be any singularity, or any difficulty reaching r=0, with a finite negative mass of nonzero volume.
Negative mass has a lot of other problems though. Thermodynamically, for instance, it's a real mess. One would expect particles with negative mass to gain negative energy from their surroundings, for instance, heating up the surroundings while the negative mass particles gain "negative energy".