Arthur (Joseph Gordon-Levitt)’s Timeline and Summary

Arthur and Cobb are hired to extract information from Saito. They fail but Saito makes them another offer: they're going to be incepting a poor little rich boy.
Arthur isn't happy about it but goes along for the ride anyway.
He helps Ariadne learn about dreaming and architecture while Cobb assembles the team.
Arthur strategizes about how to incept Fischer with the rest of them until their opportunity arises to put their plan into action.
Arthur drives his way out of a tough spot in the first dream level. Apparently it was his job to do the background check and he didn't catch that Fischer had extraction prevention training… oops.
Now they're in a hurry so they rush into the second level of the dream, which is Arthur's dream. Just like in the first dream with Saito, Arthur is a man with exquisite taste.
When the rest of the team jumps into the third level to perform the inception, Arthur stays behind and leads Fischer's subconscious on a merry chase to protect his sleeping buds.
He gets to have some fun in zero gravity and then has to figure out a way to "kick" everyone with no gravity.
He ends up using some explosives and an elevator… which seems to do the trick.
The next minute he's waking up on a plane a much richer man.