is there a way to sync iPhone with broken screen and no passcode?

Yes,connect your phone to itunes and click on device and click backup, so the backup will store on your computer.if and only iff the phone has connected to this computer previously,if it is a new computer then it is not possible because you must allow access on your phone,After getting your phone repaired you can restore that backup from computer.🙂

If you've never connected your phone to this Mac, no. You won't be able to "Trust this Computer", thus you won't be able to backup/sync.