If you still have his cup that he came with you can try to lure him in there and then scoop him out (I use food to get my guy in the cup). You just want to make sure you are doing this delicately so you don't damage any of his/her fins.
As for cleaning the tank itself, I would just rinse everything in running water (I'm assuming you don't have a filter and the tank is not cycled). Make sure you stir the gravel/substrate really good so you remove all the waste.
After you clean, add in conditioned water (water that has been treated to remove any chlorine from the tap) and then slowly acclimate the fish back into the tank (float the cup for 10-15mins so the temp will be about the same). Release the fish and repeat the process every other day.
I think cycling a one gallon would be a little hard to do...in the smaller tanks, I just personally prefer leaving them uncycled and cleaning the whole thing out altogether.
For my 100% changes (which you'll need to do if you hve an uncycled tank), I cup the betta, then use a baster to stir up the gravel and remove the poop. Then I lug the whole thing to the bathroom, dump the dirty water, and wash everything in hot water. I scrub anything that needs scrubbing with a spare toothbrush.
Pretty simple, shouldn't take more than 10 minutes.