Is a 1/3 of an acre big enough for my horse...?

It's big enough for him to get out and stretch his legs but don't count on it being good for grazing. 1/3 an acre just isn't enough land to graze on even for just one horse and if there is gr in there now it will be gone quickly. You are better off feeding hay on the ground (in a clean, dry area free of manure) instead of from a hay net. Horses can get a leg caught in them which can result in serious injuries. If you do have to feed off the ground a hay bag is much safer.