Can I download and sync music onto my iPhone from the iPad Pro?

No. iPads don't support direct syncing with other iOS devices.

You can download purchased content from the iTunes store directly on other iOS devices using the same Apple ID: Redownload apps, music, movies, TV shows, and books - Apple Support

For music previously synced from a computer to the iPad, you will need to also sync the iPhone to the computer.

If you don't have a computer anymore, then Apple Music or iTunes Match will not do anything since the source of the music for either of those must be a computer with iTunes.

Hello,

Unless you subscribe to Apple Music or iTunes Match then you will have to synchronize your iPhone with a Macintosh or Windows PC.