How to back up music on iphone to icloud

You cannot.

If the music was purchased from the iTunes Store, it can be re-downloaded from there.

If it was synced from a computer, then it won't get backed up, and and should be synced from the computer again.

Apple Music only matches or uploads music from a computer, it does nothing for music already on an iOS device.

If you need to recover your music library from your iPhone, see here: link-> Recover your iTunes library from your iPo… - Apple Community

I don't know of any way unless you subscribe to Apple Music. If not, you can always download your music purchases:

Redownload music - Apple Support

or they should be on the computer you used to sync the iPhone from originally.