Adding the old water won't do anything to kick-start your cycling. There won't be many (or any really) bacteria in the water, so its just old water that is probably high in nitrates (which won't kill them right off, but aren't good). One trick you can use is to take some gravel (the more the better) out of your 20 gallon and either add it right to the new 10 as part of the gravel there or put it in an old nylon and tie it off. There will be plenty of bacteria in the gravel that will help with the cycle. Also, if you use sponges for filter media or something else removeable like that, you can take some out of the established filter and use it in whatever filter you have for the 10. Most of your bacteria will be in the gravel though, so definitely go with that.
The only real way to kick-start a cycle so the tank will be ready to use right now is to buy some Bio-Spira and pray that is works (you get dud bags sometimes). It'll be 13 bucks for the 1oz packet, and they can be hard to find (chains don't carry it). Still, if what you want is a tank that is ready for fish now, thats the way to go.