Broadly speaking, tea plantation tourism is not a thing in Japan, for multiple reasons: the picking season is short (as little as ten days, although Kagoshima squeezes in 4-5 harvests per year), they don't want tourists in the way of the giant lumbering machines that do the bulk of the harvesting, and traditionally the places that process, roast, package and sell tea are separate from those that actually grow it.
But! After way too much searching, I found one group in Kagoshima that organizes tea tours: Chajukai (茶寿会), who in season run "green tourism" tours every couple of days that tour a live tea plantation and sample some fresh produce, for a token ¥200 per head. Two major catches: you need a group of at least 15 people, and you will likely need a Japanese speaker to book and accompany you.
If this sounds like too much hassle, you can pay a visit to Mt. Ōno-dake (大野岳), from whose viewing platform at the 466m summit you get a panoramic view of all the tea fields around you.