You will need to pull the plastic plug socket, where the new supplied power cord attaches to the new board, off. It should come off with pliers.
You will then need to snip the plug pins close to the board or it will not fit in the enclosure.
Then just solder the wires from the MacBook Air Superdrive to the snipped pins in this sequence:
RED
GREEN
WHITE
BLACK (Two Wires)
There are a total of 5 wires and 4 solder points. Both black wires are ground (and are connected together), the red wire is +5V, the green and white wires are the USB data pair.
After soldering put some Kapton Tape over the solder points.
The screws Apple used to secure their board to the DVD are too short to use with the new board, so use some additional Kapton Tape to secure the board to the DVD.
Check the pull apart pictures here:
http://www.jacobpierce.com/blog/2008/06/01/tear-down-external-usb-macbook-air-superdrive/
And tnkgrl's pictures here:
http://www.flickr.com/photos/tnkgrl/sets/72157605790040071/
You should be good to go.
Let us know your results when you finish.