Let's do some caculating.If you don't waste any time completing the ButtPass requests, you open the Legendary Room on your first day and open the Mythic Room on your second day.
You will defeat the event monsters about (60+30+15)+(60+30+15+7.5)+(60+30+15+7.5+3)×5+5×5=105+112.5+115.5×5+5×5=820 times.
And it takes (1+17)×5×17/2=45×17=765 times to upgrade the ButtPass to level 18.
So if you want to upgrade the ButtPass to level 18 without money, it's possible. But you have to defeat the event monsters almost day and night.