Q: Reduce iCloud storage capacity and put content back onto phone

If you have Apple Music, it is stored in the iCloud Music Library, but you should also have it in your iTunes Library, or you can manually redownload it to the same Apple ID:

Download past purchases

Contacts are stored in iCloud, not on your device, so those would remain in place.

Sounds like for Photos, you have iCloud Photo Library turned on? If you do, and you have Optimized your Photos on the device, you can download your originals before your storage is downgraded: Settings>iCloud>Photos, check "Download and Keep Originals".

To downgrade your storage:

iCloud storage upgrades and downgrades - Apple Support

Best of luck,

GB

Are you signed onto Settings/iCloud with the same Apple ID as you used on the old phone? Have you set up syncing?