I highly recommend cycling the tank--it will provide the fish with a more stable environment and save you a ton of work. :) Even though a betta is only one little fish, the waste they produce builds up really quickly, so you should have no trouble with the ammonia supply for the tank if you choose to cycle using the fish-in method. Personally, I don't think it's practical to attempt a fish-in (or fishless for that matter) cycle unless you have the appropriate testing equipment, I don't feel that it's an option, since you won't know when to make the appropriate water changes and how much needs to be changed. You also won't be able to tell if the tank is cycled unless you take samples of your water to the fish store regularly to be tested.
I always advocate fishless cycling, it's the method I prefer because it takes less effort because it requires less water changes, it's kinder on the fish, and it's faster because you can alter the conditions in the tank to suit the growth of bacteria without worrying about its effect on the fish. If you don't have a test kit and would rather take your water to be tested at a pet store a couple of times a week instead, fishless cycling would be the better option for sure.
Just some things to think about. :)