One of the things that worries me about negative mass is how it responds to force. If you poke a negative mass object, won't it move in the direction of your finger? Wouldn't that make an even stronger force (since the object is moving toward the force)? Is there a limit to this or would a positive-mass object and a negative-mass object be permanently attracted to each other until something breaks at a fundamental level?
Take a ferromagnetic negative-mass object of the same mass as your spaceship. Suspend it within your spaceship with electromagnets around it. To accelerate forward, try to pull the negative-mass object backwards, and vice versa. You can do the same thing if it has an innate magnetic field, or with electrostatics. (You can also do the same thing with gravity, but your acceleration will be annoyingly small)
Voila, a quote-unquote reactionless drive! (It isn't actually, the net momentum/kinetic energy of the system does not change)
But yes, there are a number of problems associated with negative mass.