How can I copy MP3s from PC to iPhone (which isn't currently synced with the PC)? I have an iPhone 4 (with iOS 6.x on it) that is not synced with any computer.

How can I copy MP3s from a PC to my iPhone? iTunes is not synced with the phone so I cannot put the MP3s into the library and sync.
The Music app needs to sync with iTunes on desktop. You can use any of the  file manager apps for iOS like Documents, transfer the MP3 files to your iPhone. The only limitation being that you'll be restricted to listening to them in the same (file manager) app and not with the Apple's Music app.