You have to beat the specified time on every level from Adventure Mode. You can see it easily by looking at their colours. A blue sphere means you haven´t completed the level in the given time. A red sphere means you have done it.
First levels are harder, because you have to little time (50 seconds for the first level, which has 5 boards), so don´t stress too much if you can´t get them on your first try. Later ones are far easier, and you can beat the time almost every time.
The first useful tip I have is how you should play to beat levels fastly. I know many won´t like it, but you can´t use the controls as you wish:
* Don´t ever use the D-Pad. It moves slower than the analog, and that means more time for the same movement. You WILL HAVE to use the analog.
* Don´t keep A pressed to select the whole rectangle. That´s slower too than just pressing A on one corner, move to the other, and press A again.
With those changes you should improve a lot, I would go as far as to say that you will end the same levels on half the time you used to.
You will need to choose the correct objects too. At one time in Adventure Mode, you will be able to choose two of them. I HIGHLY recommend the Target Shrinker Sundae. Less target = Less time. Use it with the Angel Gift Tea and your requirements will be cut by half. Don´t waste an object using something like Time Herb Cookie or Time-Pop. You want to be fast, you shouldn´t even die.
You can make levels easier by making secrets too. If you keep deleting a colour from the board, you will have 1 colour less, so it will be easier to spot combos.
Lastly, the first time you complete Adventure Mode, it will reset and you will have to start from the first level again, but it will keep your objects, characters, records, etc. If you finish it the second time, it won´t reset and you will be able to choose any level and end the game as many times as you wish. You must end it twice to unlock this achievement.
The achievement pops as soon as you beat the last time you hadn´t beaten yet.
Really thanks to JotaroRaido for his comments!