"Животноводство" (from "животные") describes animal husbandry, particularly cattle-raising. From how long ago is this word attested?
Well, these kinds of terms — I mean terms for describing some branch of economical activity as a whole — usually came into active usage when economical statistics became a thing.
In Russia (more or less like in the rest of the Europe), this happened in the second half of the 18th century. One of the first economic entities in the Russian Empire was (to the best of my knowledge) the "Free Economic Society" (in Russian, "Императорское Вольное экономическое общество"), which published a magazine dedicated to descriptive economic statistics.
In this magazine, you won't find "животноводство". However you can easily find "скотоводство". This word has been used since the very beginning, as early as in the 1765 issue, as you can find on page xx 4:
Дѣлать вѣрные опыты, касающїеся до домостроительства, земледѣлїя, береженїя и размноженїя лѣсовъ, скотоводства, рыбныхъ и звѣриныхъ промысловъ, горныхъ дѣлъ, мануфактуръ, всякикъ рукодѣлїй и прочаго, [...]
Technically, the term "скотоводство" is narrower than "животноводство", since it only refers to livestock, but the de-facto term "скотоводство" was widely used throughout the 19th century (and it's actually still widely used) while "животноводство" gradually became more frequently used toward the end of the 19th to the beginning of the 20th century:
There's an article in the famous "Энциклопедический Словарь Ф.А.Брокгауза и И.А.Ефрона" about "Зоотехния" (there was an attempt to use this word as a more scientific term, but it never became popular — even though "зоотехник" is still used), in this article we read:
У нас эти названия переводят словом "животноводство", а несколько десятков лет тому назад, нынешняя З. называлась просто скотоводством (Viehzucht).