"Sukiyaki" is a term with absolutely no relevance to the song's lyrics, as sukiyaki is a Japanese beef dish. It was used in the song name only because it was short, catchy, and recognizably Japanese.
The original title "上を向いて歩こう" means "I Look Up When I Walk". The lyrics tell the story of a man who looks up and whistles while he is walking so that his tears will not fall.
This is the only Japanese song to hit #1 in the United States. Moreover, it is one of the best-selling singles of all time, having sold over 13 million copies worldwide.