To the left of the lift you will see a crate, drag it to the right so the lift will pick it up. Activate the lower switch and ride the lift up with the crate. At the top, push it off onto the upper platform.
Note: Hit the switch to lower the platform again. Now, use the switch to send the lift up again, but use the crate to jump on top of the lift. Ride the lift up and as you near the top, look to the left and you should see a rope hanging off to the side. Jump over to this and an egg will pop out of a pipe higher up on the wall. Jump back to the lift and drop down to the ground level. Cruise to the right to pick up the egg for the achievement/trophy - Urban Exploration.
Use the elevator to get back up to the upper platform with the crate. If you move a little to the right you will notice a see-saw type platform that will hinder your progress. You will need to stand on the near end so that the platform is sloping upwards to the left. Make sure the lift is at the very top of the room. If it is, hit the switch, then grab the platform and quickly pull it to the right. Place the lift platform on the nearest end of the see-saw. Now you can freely move the crate onwards to the right.
Pull the lever near the far wall, then position the crate below the rope when it lowers. Pull the switch one more time to send the rope back up and use the crate to quickly jump onto it. Ride the rope to the top and swing over to the platform on the right.