"...I can assure you that the first day you will get to unlock the mythical girl..."
surely Colosseum is a great help, but 1st day? But, lets calculate:
quest 10 Find 125 event common cards - common CD 24min (25cards) =4 fights (1 free)= 1h40min, or 4 common dolls
quest 13 Defeat rare monsters in the Dungeon rooms 20 times - CD 50 min = 3 passes (1 free) = 2h30m or 15 rare dolls
quest 15 Open any 3 chests in the Shop - (1 free) 2 keys or common chest CD 4 hours*2=8 hours
quest 17 Find 160 event rare cards - rare CD 48min (20 cards) - 7 fights (1 free) = 5h30m or 7rare dolls
quest 18 Defeat epic monsters in the Dungeon rooms 20 times - CD 1h30m = 3 passes (1free)=4h30m or 15 epic dolls
quest 22 Open any 4 chests in the Shop - (1 free) 3 keys or CD 4h*3=12hours
At that point, if you're new player and don't have Colosseum boosters you need 1h40m+2h30m+8h+5h30m+4h30m+12h=34h10min. And you not even close to leg monster.
But even if you have Colosseum bonuses, you need at least 4 common dolls, 7 rare dolls, 15 epic dolls, 5 keys. Which is more then standard yield from Colosseum or about the same if you minus Epic dolls.