How to get music from my iphone 5 onto itunes?

My Iphone 5 screen cracked and it is being replaced by a brand new Iphone. In the apple store I was told I would have to download everything from my phone into itunes so when I get the new phone I can sync it back. However, I don^t know how to get music from my PHONE to ITUNES because whenever I try to sync it, it tells me music on my phone will vanish since I have NOTHING on my itunes.
I have already tried:
-backing up to icloud
-Syncing
-have googled many options

I would not like to:
-purchase itunes match or extensive icloud
-purchase or download anything on my computer or phone
-ONLY replace the screen on my iphone, I prefer getting a new one

Thank you for your help, serious answers only please.

Added (1). I did not purchase the songs from itunes. Therefore, I can't go to my purchases and sync them. I would prefer not having to redownload all the songs. However, I have purchased some apps. How can I make sure they get onto my new phone aswell? Are my apps already saved and ready to sync?

My Iphone 5 screen cracked and it is being replaced by a brand new Iphone. In the apple store I was told I would have to download everything from my phone into itunes so when I get the new phone I can sync it back. However, I don^t know how to get music from my PHONE to ITUNES because whenever I try to sync it, it tells me music on my phone will vanish since I have NOTHING on my itunes.
I have already tried