The Imperial Palace loop is the obvious starting point — 5km, flat, well-lit, and busy with locals from about 6am. It's a genuinely social run too; nod to enough people and you'll find yourself unofficially pacing alongside a stranger for a kilometre or two.
Beyond the Palace loop
If you're staying in Kyoto for a stretch, the Kamo River path is the equivalent — flat, scenic, and popular enough that you never feel unsafe running before sunrise. Both routes reward an early alarm far more than they punish it.