Just go to iTunes on the second computer and right-click your device and select "transfer purchases" to move your iPhone apps to iTunes. As for music, not sure how if it isn't in the library. The only thing I can think of is to save your music files from the first computer to a flash drive and transfer them into the other computer.
I could be wrong but I don't think there is an app that will allow what you're looking for as most everything regarding syncing runs through iTunes and erases anything on the phone that isn't on the computer itself when syncing.