Without proper context, I'm going to assume "this guy" is Clan Boss? 1M damage in one day will eventually become absolutely effortless for you. That's the amount of damage I can do in one turn. The fact that you are struggling to reach that amount over the course of the fight likely means your understanding of the mechanics of the fight are the root cause of your problems. Why don't you post your roster / team that you use against him? Perhaps we can help clarify some things.
As for when the day starts - there are two moments, actually. At 8PM EST / Midnight UTC, the quests reset, along with most other content (Doom Tower, Faction Wars, etc...). At something around 6AM EST / 10AM UTC (I'm not sure the exact time since I'm asleep usually), Clan Boss resets. Around that same time, weekly arena resets occur.