Looking at an Ngram of the phrase I noticed a sharp rise in its print frequency starting just before 1940. Scanning references from that time period, I found numerous mentions of Golden Boy, a commercially successful 1937 play by Clifford Odet (inspiration for the Coens' Barton Fink). This is also the earliest reference given by the OED as quoted by @Cerberus in his answer to the linked queston, On being golden.
I then found this definition in Brewer's Dictionary of Modern Phrase & Fable that confirms the influence of Odet's title:
Golden boy or girl.
A popular or successful person, especially in sport or business. In the former, it is usually implicitly connected with one who wins gold medals, especially when handsome or attractive. Thus the good-looking US boxer Oscar De La Hoya was dubbed the 'Golden Boy of Boxing' after winning the gold medal in the 1992 Olympics. In Clifford Odet's play Golden Boy (1937) the hero, a violinist, is also a successful boxer.
Other Google Books listings confirm the use of the phrase as a favorite in sports lingo, with various countries and sports having their own golden boys.
The phrase certainly predates this popularization, though. I found several figurative uses of the phrase from the 19th century, notably this 1848 reference describing a character from Goethe's 1773 Goetz von Berlichingen:
Then there is George— " the golden boy," the joyous and lighthearted aspirant to chivalry, whom old Gotz loved as a part of himself, and who is indeed the very perfection of boys.
The phrase is in quotes because it is being used as a direct (translated) quotation from the play. Sir Walter Scott's English translation of this play has "gallant boy." If any German-fluent users here could confirm that "golden" is a more accurate translation, then this may be the first example of its modern connotation.
In other news, I found early references to the Japanese legend of Kintarō whose name is often translated "Golden Boy" and is the inspiration for the anime title mentioned by the OP. This popular Japanese folk hero, a child of superhuman strength, could also be the origin of this phrase in English though I couldn't find translations of his name as such before this 1896 reference:
The hero of Japanese boys is Kintarō, the "Wild Baby," the "Golden Darling." Companionless he played with the animals, put his arm around their necks, and rode upon their backs. Of him we are told,"He was prince of the forest; the rabbits, wild boars, squirrels and pheasants and hawks, were his servants and messengers." He is the apotheosis of the child in Japan.
Also of note, while unrelated to its origin, is the use of the phrase golden boy in gay subculture since the 1970s to refer to a young man in his prime.