Welcome, sorry your new guy isn't doing so well..
Distilled and filtered and boiled water is not ideal to use in aquariums.. it lacks the necessary minerals. The best water to use is tap water (or if you have a well at your house) with water conditioner. Anything else is not beneficial to them.
So go ahead and fill up the new tank with tap water, use your glass aquarium thermometer to make sure it is roughly 78*F, and add in the appropriate amount of water conditioner.. float him in it for roughly 15 minutes, during that time take out a bit of the water from the cup (toss it away) and replace with water from the tank.. I suggest doing that twice at least.. this way he become acclimated to both the temp and the new chemistry.
Air pump is not needed, as bettas breathe using a labyrinth organ, and breathe the air we do.. the aeration can be too much for him in that small of a tank, and cause him to stress out.
The heater should be fine- when doing water changes, make sure to get the temp as close to 78 as possible so the heater won't have to work so hard heating the water back up.
Weekly water changes of 1 50% and 1 100% per week is needed in an unfiltered tank that is below 5 gallons. Use water conditioner each time with only tap water.
Feed daily, small multiple times a day.
Not to be mean, but goldfish should live for over 20 years.. so living just a year is not a good thing. Goldfish require 20 gallons minimum for a single goldfish, along with dual filtration... A betta has a life span of 3-5 years approximately.
Make sure in his tank you have silk plants to avoid ripping his fins.. a cave is also loved by males. A 1 gallon is a fine size for a single betta, just have to keep up on the water changes.
Otherwise.. get him in the appropriate water conditions, make sure the temp is 78* and he should come around. If not, it could be the stress of the move and new home, along with the stress of the different water being used if he wasn't acclimated properly.
Good luck to you..