The story
By proprialization from rising Sun, driven by the theme that the Sun is very important in human affairs and that sunrise is an important time of day, both for the advent of daylight and because the rest of the day, with all its potential, lies ahead. The Japanese symbolism reflects that Japan is the easternmost country of East Asia, and the East is the place of the sunrise. The pub names reflect both the general importance of the rising Sun in human affairs and also the heraldic use of the rising Sun as a symbol for royalty, nobles, goodness, and so on.
Etymology from Wiktionary, CC BY-SA 4.0.