Pre-Meiji Japan was never called the Sick Man of Asia. That particular phrase was coined by the British press for China in the late nineteenth century, decades after the Meiji Restoration.

British newspapers began to use the phrase "the sick man of the Orient" to describe China in 1896. Other newspapers followed suit. Several variations of the phrase, such as "the sick man of Huaxia," and "the sick man of East Asia," were in circulation at the beginning of the twentieth century according to Gao Cui (see his book, From "The Sick Man of East Asia" to A Strong Country of Sports [Cong "dong ya bing fu" dao ti yu qiang guo] (Sichuan renmin chubanshe, 2002), 9-12. In 1905, The New York Times (16 August 1905, 6) reported under the title "The Sick Man of the Far East": "China will henceforth be put in the same position in which Turkey has been for half a century.... She will become, by the ratification of the proposed treaty of peace, the Sick Man of the Far East." Western scholars usually use the phrase "the sick man of Asia."

• p. 22, Public Health and the Modernization of China, 1865–2015 by Liping Bu

I hope some of our Chinese history flairs can give you more background on how this phrase was employed re: China.