Welcome to the forums. :)
Unfortunately, you'll find many here who will tell you that cycling in a 2.5 gallon tank is impossible or at the least nearly impossible. It's not worth trying. However! You'll have to do less water changes than in a smaller tank, so that's a plus! I'll let someone else recommend how often to do them but cycling just isn't going to happen for you. :/ To "cycle" you need a tank 5 gallons or larger.
While you've got him in the little tub he came in though, you'll -definitely- need to do daily water changes for him in there. Just try to keep him somewhere warm and keep the temperature of the water you put him in the same.
There's no reason if you've got water conditioner that you couldn't introduce him into the tank you've got right away. Start by just floating his little cup in it for about 15 minutes. Put a handful of the new water in his cup and keep doing this in 15 minute intervals for an hour or so. He'll adjust to the water and be ready to go in real soon.
I don't know what kind of filter you've got. If it's one made for the tank, it'll probably be fine to run as it is. Stronger filters need 'baffling' as bettas don't like rough currents. Simple process of sticking a sponge or something on to slow it down. There's a guide
that has a more involved process that could be fun for your 6 year old to try too!
He should be fine for a couple days without the heater but make sure you try to keep him warm.
Good luck and again, welcome! He sounds like a pretty fish and I hope you share some pictures of him. :)