if he really is in a bad state, so you think hed pass peacefully.
there is really no method of euthanizing a fish that is too peaceful.
keep changing his water, offering food, etc., and just let nature take its course.
if i were a fish, and dying from old age (and really a year isnt all that old to own a betta, even if already a year old when purchased), i would want to die peacefully, rather than be "helped". if he doesnt look like he's suffering, just treat him as normal and let him be.
it is sad that it comes to this
all the best