I wouldn't worry too much about the pH, and I would not recommend changing the pH with any chemicals-most fish will adjust to your source water pH without problems, what is best is a stable pH as pH swings can be deadly.
Bottled water is not regulated and often it is just tap water that has been filter and can still contain chlorine and chloramines that are harmful to fish...always use a good dechlorinator
Also, due to the filtering process of the bottled water it often will have the needed minerals filter out that the fish needs for good health and this needs to be replaced, there are products that you can buy to add this back or use a 50/50 mix of bottled/tap water.
The nitrogen cycle is important, however, in 2g tanks due to the needed water changes of 50% twice week in a filtered tank, you don't have to worry about the cycling process per se, it will happen, and provided that you are not overfeeding and removing any uneaten food after feeding...water quality should be fine.
You want to keep ammonia and nitrite 0.25ppm or less..ideally 0ppm and nitrate 5-10ppm
Water temp in the 76-80F range-
On the filter-it needs to run 24 hours, after the filter has been off for 6 hours the good bacteria start to die and convert to ammonia-when you turn the filter back on without rinsing you can dump a lot of harmful ammonia in the tank, this also can stop the cycling process and/or cause mini cycles that are stressful on the fish.