NOTICE: The description is misleading. You must complete Wind Fortress within 2 minutes!
The trick to this is to use the blade in the first section, and charge shots in the second section. Use blade on the boss, and if you think you're going to die near the boss, damaging yourself on the red laser wall for one damage is a nice trick.
This video was recorded by me. I barely made it, but all the strategy to complete this achievement is there with a general route you will want to follow. As you can see, I can be fairly slow and make a few mistakes and still finish in time. You may get a few runs around 2:20 like I did, but eventually it will come with practice. Mastering the booster is key to making it through this effectively, and after all, it all comes down to practice.