If I eat a carrot a day, how long will it take for my eyesight to improve?

Forever. Carrots contain carotenoids which are an essential component of visual light receptive pigments in the eye. Almost all diets have enough of this that deficiency is extremely rare and is found in this country only in bizarre fad diets. If you vision is suboptimal you should be evaluated by an ophthalmologist and not wait for eating carrots to improve things as that will be a long wait.