You can try to cut and root it, but it will likely not do well in the water. The Dracaena would be more apt to root by dipping the cut end in rooting hormone and then planting it in another pot with a mix of peat moss, vermiculite (sp?) and potting soil. When you cut this, be sure to cut above any brown on the branch.
As for how long it'll take to reach 1' to 2' in height...depends on how tall it is now...if it's just a small one, could take about 2 years to reach about 1 1/2' high.