How can I use a second computer to manage apps on my iPhone?
You are quite correct, you don't want to sync your iPhone to this new Mac. At the least, you'd lose any songs or movies you have loaded.

You can do what you want using the iPhone Configuration Utility (free at http://www.apple.com/support/iphone/enterprise/). Install, connect the iPhone, click on the iPhone's icon, and go to the Applications tab. It will let you install or uninstall individual apps.

Or connect your iPhone to WiFi (you could use Network Sharing from the new Mac, if no other WiFi is available) to download directly, after buying from either the iPhone or the Mac.