Purchase music on iPhone ... download on phone and computer?

"Yes,

The next time you sync your phone (if you have sync music turned on), it will transfer any new purchases to your computer. It checks for this every time you sync.

It is the same idea with the applications. When you purchase an application through the iphone, it will also sync to the computer on your next sync (if you have application syncing checked).

I personaly do this all the time and it has worked great every single time."