I wouldn't use bottle water if I were you. Use regular tap water, with a dechlorinator, (Stress Coat). Tap Water has stuff inside it that the Betta's need, and with a good dechlorinator, all the harmful stuff like Chlorine will be gone. Also you asked about Cycling in the title of the thread, but do you know what the Nitrogen cycle is?
I would research it a bit, then buy an API testing kit and by some Ammonia, (a scentless brand that when shaken doesn't make bubbles). It'll take about 3 weeks, and you should keep your fish in another tank, or a Kritter Keeper. But its much better than doing a fish-in cycle, which can be dangerous to the Betta. Also do you have a heater? Didn't know if you mentioned that or not, but Betta's need one ;) 25 Watt should do.