There's no such thing as "negative mass", at least not in this context. Antiparticles have the same mass as the corresponding particles. This has been carefully demonstrated in experiments to check whether antiparticles "fall up". They don't. They fall down, just like particles. In fact the term "antiparticle" is a bit chauvinistic, since you could just as easily build a working universe out of the opposite charges etc. In the Big Bang we presume there were equal numbers of both, but today we apparently have almost none of the "anti" type hanging around. Why hasn't all the matter and antimatter just annihilated and left nothing behind but photons? The answer is suspected to be found in the tiny "CP" asymmetry in decays of certain rare and exotic particles which favour the production of matter over antimatter by a small fraction of one percent. Weird, eh? As for making your own antimatter, all you need is a source of gamma rays with energies higher than 1.22 MeV: run them through a thin lead sheet and Voila! out come pairs of electrons and positrons. Of course, the positrons annihilate with other electrons within a few billionths of a second, but for a while there you have a bit of antimatter. Personally I wouldn't recommend hanging around a gamma source, though, unless you are very fond of genetic mutations and such. Also I wouldn't recommend piling up too much antimatter in one place, as you have to be very careful to keep it away from matter or else BOOM! There is some talk of storing antihydrogen in magnetic bottles to use as rocket fuel, but I'd be pretty nervous sitting in that rocket. (Actually I'd be pretty nervous sitting in ANY rocket, but it might be worth it.)
making a small donation to science.ca.