How to put music on iphone from different computer without erasing all music?

Got a new computer. I have the new itunes store but it won't let me put music on my iphone. Everytime i plug it in it shows my iphone name, but when i click on it, it goes away. & i really want new music on my phone. Please help. Also i don't buy my music on itunes. & i have windows 8 pc.

When you click on the icon in the upper right, don't click on the arrow pointing up. That ejects the iThing. Click on the center/left side of the icon.