A Rose in the Twilight

You can attempt Time Attack after beating Blanc at Clock Tower Upper Level and picking up the hourglass by coming back to the area. However, you can only see your completion time after defeating the final boss Viola in the Sealed Room.

When you beat the game for the first time, your completion time will most likely be much higher than 1 hour and 20 minutes. That’s no problem at all since you can attempt every level over and over again and aim for a better time. Every second you’re faster than before helps you to achieve a perfect completion time at the end.

There isn’t really a best way to do it. You need to know the levels by heart and find ways to save time. Get perfect timing in puzzles and try to find shortcuts. Carry Rose as Giant whenever it's possible and only let go of her when both need to part ways or pick up something.
After memorizing the patterns of every level and being able to perform them perfectly, you’ll get the trophy eventually.

I took some screenshots of my times. They should help you to see whether you were fast enough or still need a few more tries. I was blundering now and then, so there is definitely room for error. I messed up in Library 1 ?3 and you should be able to save around 2 minutes there.