First, what breed is the goat? If he is one of the minis or dwarf breeds he may be small to castrate. We have successfully banded up to 5 months on sheep. As long as the scrotum and testicles will fit through the band yu can do it. At 10 months you will need some strong help while getting the band on. If the testicles are too big, you will have to surgically castrate and that means a vet.
When docking week old ewe lambs with a band we give a shot of tetanus antitoxin which gives immediate immunity along with the regular CDT shot that includes tetanus protection later. We like to be sure the animal is protected instantly. Once the sheep or goat is several months old and has had it's 2 vaccinations of CDT it doesn't need the antitoxin. When banding older ram lambs, we don't bother with the antitoxin. If your buck is up to date on his shots, and his testicles will fit through the band (probably with some tugging and manipulation) go ahead and band him. As long as you make sure his urethra not in the band, and he can pee, no problem.
If they won't fit, you will need surgical castration by a vet = $$$. Oher options are to sell the buck or put him in the freezer. At 20 months the meat will still be without taint. He is still sweet and gentle now but he is not completely mature, and his attitude may change, particularly if he has been a pet. Unless you have plenty of pasture/forage/$$$ for hay, why keep a wether? Personally, I look at this from a cost vs use viewpoint, but if you want to keep him as a pet then you may want to check how much the vet will charge to castrate him and make a decision based on that.
