The salt never breaks down or evaporates. You only need to readd along with water changes, then you only need to add as much salt as water you change.
I would not do any changes for the next couple of days... let him/her get used to the new water.. the lack of acclimation shocked him and hopefully he'll recover in a couple days. Generally speaking a half gallon is not large enough for any fish.
2 gallons is minimum to keep bettas in, so your half gallon should be alternating 50% and 100% every other day, if not 100% daily.
You also can't really heat that safely, however, the bowl temp will be several degrees below room temp, so you need a heater. If you get the 2g you can get a good 25w adjustable heater like a marineland visitherm, or jager. Any heater you get must be used with an aquarium thermometer that stays in the tank. You should give any new heater a test run in similar size container and use the thermometer to monitor the temperature over 24 hours to make sure it will hold an constant (no changes) between 78-80F. After you know it's working you can acclimate your fish to warmer water no more than one degree per hour and 5 degrees per day.
Look for a good quality pellet whose first two or three ingredients are whole
fish, not fish meal or wheat. New Life Spectrum Betta has garlic in it and may entice your fish to eat. Your betta should get two small meals a day with one fast day a week - how many you feed depends largely upon the size of the pellet, but also the size of your fish.
This is how you can do water changes without shocking:
In the 50% I would suck through the gravel with a turkey baster to remove poop and half the water. I would refill it with same temp water (use tank thermometer to match running tap water) that already had conditioner added to it. I use 1 gallon water jugs like you can buy at the grocery store - rinse thoroughly before using with hot water but not soap or chems).
For the 100% he'll need to be removed. I would cup in an opaque plastic solo type cup and wrap a towel for heat. Thoroughly rinse the tank, gravel and decor but don't use soap or chems. Refill with same temp water, add conditioner, stir it up a bit then float him in the bowl an hour while slowly adding a couple tablespoons of new water to his cup every 10 minutes. Finally release, keeping as much of the old water in the cup as possible. He shouldn't show any signs of shock this way.