Scoop him out in the cup he came in, gently - do not net him, since that can tear his fins. Leave a little air space at the top and put on the lid or cover it with saran wrap with a hole or two poked in it. Betta can jump, and that's the last thing you need.
Clean the tank, replace the water with similar temperature conditioned tap water (testing temperature with the inside of your wrist is fairly accurate). Float him in the new water and acclimate him gently by blending the new water into his cup water over 20 minutes or so, just like you did when you got him, and then let him go.
It's going to be stressful for him until he gets a new tank. There's no way to avoid that. As long as you are gentle with him and take your time, he'll probably be okay until you get him better quarters. You can try putting his cup into his tank at feeding time, and giving him the pellets inside the cup, to train him not to be afraid of it.
You really do want to change the water every day. Every other day at the absolute maximum if the delay cannot be avoided.