Breakfast in Japan
Japan just doesn't have the "eat breakfast out" culture the way that , say , the USA has with diners offering breakfast almost anytime, or cafes in Europe offering coffee, breads/cheese and pastries.
A typical breakfast for the Japanese is fish with rice, pickles, natto (fermented beans) and miso soup, and traditionally these were prepared at home, so the popularity of eating such breakfasts out and about have only become more popular the past 4 or 5 decades since more and more Japanese began to live alone and demand for such conveniences grew.
That said, if you want to skip the western style eggs and toast (available at "kissaten" coffee shops as "morning set" or at fast food chains like McD's) and have fish with rice, etc. the easiest places are 24 hour chains such as Tokyo Chikarameshi or Yoshinoya, which offer a simple set "teishoku" meal of fish (usually salmon), rice, pickles and miso soup for around 600-700 yen.
Either that or wait a couple of hours until most restaurants open for lunch....!