It will still be a while before this is directly observed.
We're fairly sure that it "falls" downwards for a wide range of theoretical reasons and also due to the 1987 observation of gravitational lensing in neutrinos and antineutrinos from a supernova. If we observe the opposite in a lab then it will be incredible because we'll have to reevaluate a lot of what we think we know about physics. If we observe it "falling down" then it will be an important result but not at all revolutionary.
Antimatter has opposite electrical charge (and magnetic moment) from normal matter. It should have the exact same mass and interaction with gravity as normal matter.
An anti-electron is called a positron because it has a positive charge instead of negative. Antiproton (sometimes called negatron, but rarely) is just a proton with opposite charge. You can even make a hydrogen atom out of an antiproton and a positron, called antihydrogen. https://secure.wikimedia.org/wikipedia/en/wiki/Antihydrogen
I meant falling downwards towards the surface of the earth which would experimentally show that there is an attractive force between matter and antimatter. Experimentally showing that antimatter gravitationally attracts antimatter is next to impossible for technical reasons but there is even less doubt about this than about matter attracting antimatter. No matter how sure we think we are about something we can also be wrong though. Experiment gives us the final word (albeit with statistical limitations) and there have been numerous times in history where established theories have been flipped upside down by experiments.
We're fairly sure that it "falls" downwards for a wide range of theoretical reasons and also due to the 1987 observation of gravitational lensing in neutrinos and antineutrinos from a supernova. If we observe the opposite in a lab then it will be incredible because we'll have to reevaluate a lot of what we think we know about physics. If we observe it "falling down" then it will be an important result but not at all revolutionary.