It will be easier to treat him in the 1/2 gallon since you should be changing the water every day...
1: I would move him to the hospital tank and continue treatment, changing water every day and adding the med or aquarium salt so water always has proper dose in it...the clean water alone will be beneficial to his healing.
2: get your heater (make sure it is adjustable...I go for whatever heater is cheapest as long as it's adjustable), filter (don't waste money on "bio-wheels" as study I read shows no discernable improvement. Since you are new to this, go simple and get a Hang on Back (HOB) of tank filter. It is OK if there is a bio-wheel, but its not necessary. Whispers, aquaclear, and penguin are all ok...they all function about same, I like any filter that has a sponge for collecting more of your good bacteria...usually in addition to filter cartridge), and accessories as soon as possible.
3: please read the sticky on this site under betta housing re: CYCLING your tank. It is why I say put betta in old home right now....before you add any fish, shrimp, snails to your tank, itneeds to be cycled which means that you set tank up with gravel, filter, heater, a cave (i use glasses or candle holders, etc) for your betta to hide in and relax, and plants and let it run for quite a while (longer than a week in most cases) in order for the good nitrifying bacteria to become established...this ensures that your fish waste such as ammonia will be broken down into less harmful molecules. This makes it possible to do less water changes!
4: PLANTS....I almost always suggest live plants or a combination live and silk (you can get these at a craft store or walmart to save $ and give you more variety...just make sure you choose silk (synthetic) flowers/plants that have no glue, glitter, et,). If you are going to have a community w/different species such as betta and neons, you will want many plants, and maybe some cool rocks or driftwood, to break up line of sight and make natural areas where the different species can carve out a territory.
Live plants: are great, natural, will help by absorbing ammonia (sometimes called the silent killer!) and other toxic substances from the water. If you go for live plants, you will need adequate lighting from your hood...minimum of 6500watts (if your hood has sockets, you can get those fluorescent "coiled" energy saver bulbs at pet store or walmart (with other light bulbs, not pet section)...look for "daylight" bulbs. This should be enough to grow most "low light" plants such as java fern, anubias, some cryptocorynes...try to get a variety of these. Most importantly, get some stem or bunch plants such cabomba, ludwigia, hornwort, water sprite, wisteria...these grow fast once they're rooted in the gravel...because they grow so fast, they have increased uptake of ammonia, etc, which makes for a healthier tank and fish.
5: do not buy neons until tank is cycled!