I'd be surprised if 6 days was possible, even with lots of luck. You'd need to get two workers on day 1, get a deep mine for copper by day 4, and have enough rock for 2x strength upgrades and several energy upgrades. I just don't think it's possible even with luck on your side.
You have to be really lucky about the spawns and ALSO about the miners stacking together. If on day 6 a few fall down the same hole, they will be more efficient (spend less energy and go further down) so if you don't buy too many miners and focus on maxing strength and energy and coffee, maybe... just maybe... you could reach level 300.
Also your path in the last day shouldn't have many resources and mostly rock. Because resources tend to distract miners and force them to move sideways, they should optimize by being mostly in the same hole.
I think (not 100% sure) you could technically reach it on 6 if you get super lucky and have enough upgrades and coffee, the numbers are there, but you'll have to play a lot. It's not worth it, LOL, you are the first one to reach the cave on day 7, so you're the king now!
Also, I'll update the game in a few days, so all this will be changed so don't bother.
Still, great achievement reaching it on day 7!