I think unless you do regular 100% water changes, it'll eventually cycle anyways. you can do a fish in cycle, just have to make sure to be keeping up on water changes. generally 2 50% a week if you're doing a fish in cycle, then 1 50% week once it's cycled should be good
1. You should go through the cycling process because if you DON'T it will do it anyway. It will likely do it when you least expect it and kill your fish.
2. While cycling no more than 10% per day. (more info below)
The nitrogen cycle of the conversion of Ammonia into Nitrite then Nitrate occurs naturally, this is what the old "five to ten weeks" is about, it takes that long for the bacteria to accidentally fall into the tank then thrive.
Hagan Cycle, TLC Optimizer, TLC Start Smart are products that I've used which will reliably introduce nitrification bacteria into the tank. Several other companies also carry the same material however API's "Stress Zyme" is a tank health bacteria instead of a cycling bacteria.
While you don't "need" to have an air pump and stone for bettas you kinda do need it to make the cycling work better. I find air stones for high aeration of the water never hurt the tank or fish(s) health. Easiest way to deploy the air stone is inside the back of the hang-on filter. It will make some popcorn noises or sizzling bacon noises depending on filter make.
I don't like Aqueon 5gal tank kits simply because the Aqueon filter is trash. Walmart's 5gal kits don't have a heater but do have a marineland manufactured filter which uses a standard filter card and a bio-fiber insert.
If you can afford it an assortment of an Aquaclear 20, an AllGlass 5gal with lid and a Penn-Plax 5gal "cascade" heater should rock the world for you. The Aquaclear 20 can be turned waaaay down in flow rate, in my opinion it is the superior 5 gallon filter for betta keeping.
If you "worry" about cycling later you may still harm your fish. Best to start it off right at the beginning.
When you get the 5gal tank don't put in any gravel/rock/glass/dirt, just put in the ornaments you will use and set up the filter with the card in it. Run it for a day or so without fish, just treated water and filtration bringing the water to temp to age it and let the carbon filter clean all the 'whatever' off the ornaments.
Then check your water parameters, insert fish and feed him immediately so he knows it's permanent. Feed him normally for a couple days with the filter card in the filter then second or third day remove the carbon filter card and pour in the first dose of cycling bacteria with the light off for the next five hours. This will allow the first bloom to occur and give it a chance to attach to the bio-media. Feed him regular for several days and let the bottom get dirty if it does. Second dose of bacteria and a few more days and you should see the cycle operating, ammonia nitrite and nitrate. If you don't see nitrite but you DO see nitrate and ammonia, that is fine. If the ammonia gets over 1.0 then you need to change the water out to whatever ratio takes it back below 1.0. If the nitrite reaches 1.0 do the same. If the nitrate passes 30 do enough to take it back under 15 or so.
After a week you should have enough biological filtration established and can go ahead and clean off the inside walls of the tank and waft water across the ornaments. Siphon sweep the bottom of the tank (fold the hose so you don't waste water, that way you can start and stop the flow without losing siphon), put the filter card back in and you can add up to a half inch of rocks if you are not going fully planted or you can add plants at this point.
2. Water changes.
Uncycled: Turn off water filter and let the water rest for ten minutes to stop flow so that debris settle. Take filter off back of tank and rinse out with hot tap water then let rest to dry. Siphon 30% of water out through rocks on bottom using small circular motions every two days. On the sixth day you do a 100% water change including rinsing off and bottle-brushing ornaments and tumbling rocks (big food-service mayo bottles are good for tumbling rocks) in salted water. Use one tablespoon aquarium salt per half gallon when cleaning the rocks. Rinse out everything, return everything to the tank EXCEPT FISH then fill with treated water and run for about half an hour before putting fish back in.
Cycled: 25% of the water siphoned through the bottom rocks every three days and 50% water change on Saturday should keep it all very clean. Turn off the filter when you scrape algae off the side and clean the filter about once every two carbon card changes but do not clean the bio-media components.
Remember that any time you mess with anything in the tank you should turn off the aquarium heater.
Hoshi will appreciate the space. Cycling isn't that hard, it just sounds hard cos of all the chemical names and bacteria and water changes. But once its cycled, water changes will be far fewer. Good luck!