Arcade mode is the same as story mode, but with scoring and a time limit.
You unlock the arcade mode version of a level after beating that level's boss. To access arcade mode, select Single Player from the main menu, then press right.
The easiest level is the first one, the Jungle. You'll have plenty of time to complete the level. Rush through the level as fast as possible, but if you're dying a lot, slow down a little.
When I did this, I died several times and still had six minutes left, so don't fret if you die.
If, while playing the story, you didn't get:
The Untouchable achievement in Outland worth 49 pointsDefeat any boss without taking damage (General)
Then you should get it at the same time as this.