In the lower right-hand corner of the map, the mapmaker describes the map. With the help of Chaamiey (See comments section), the following is my translation of that description:
This map is a reduced, modified version of a map made by Ino Tadataka (伊能忠敬) in that it has omitted Hokkaido and added Korea and the East China Sea. Traditionally, we have referenced Inō's map when drawing our country's coastal regions, but the surveying technology of his time was still imprecise, so there were frequent errors and omissions. Therefore, even though this map is based on Inō's original map, our navy has corrected the surveying errors of the coastal regions. For areas our navy has yet to survey, the results of Western surveyors were used as a stopgap to reduce suspicions our navy has with the original map. The navy intends to make furture modifications as soon as new surveys are available. This map is not perfect, but we hope it is of some help to those navigators who reference it.(See the Comments Section for a transcription and Japanese translation of the map's old Japanese description.)
11th Month of the 11th Year of Meiji (1878)
Japanese Navy Waterways Bureau
As you can see from the description of the map, Western maps were referenced for areas the Japanese navy had not yet surveyed, which almost certainly included Ulleungdo and Liancourt Rocks since the Japanese survey ship Amagi did not visit Ulleungdo until 1880 (See HERE). In fact, if you look at the comparison below with the 1876 British Navy map, you can see that the Japanese navy most likely got their surveys of Ulleungdo, Liancourt Rocks, and the Korean peninsula from the 1876 British map, which also does not show the non-existent island named Argonaut.
Japan's Council of State, the Dajokan (太政官), did not instruct the Japanese navy to survey Korean coastal waters until April 1878, so those results are almost certainly not reflected in this map.
If you compare the above map section with the following section of an 1876 Bristish Navy map, you will notice that the sea depth readings and shapes of the islands are the same, which suggests that the Japanese showed Ulleungdo, Liancourt Rocks, and the Korean peninsula on their map by borrowing from the 1876 Bristish map. The British also labeled Ulleungdo as "Matsushima" and Liancourt Rocks as "Liancourt Rocks" and also omitted the non-existent island of Argonaut.
The following is the description found in the right-hand corner of the Japanese map.
You can view the map in more detail HERE.