Q: sync 4s with black screen to my new 6s

If your contacts were synced to iCloud or another email provider - installing that email account will download those contacts to your phone

On the old phone -

If the screen on your iPhone, iPad, or iPod touch doesn't respond to touch - Apple Support

If that does not work AND you have a computer with iTunes to which you have previously backed up the 4S - trying to restore it from there will prompt a backup before the erase

If you can get that to work - you can then restore the backup to the new phone which will transfer all your content

How to erase your iOS device and then set it up as a new device or restore it from backups - Apple Support

Transfer content from an iPhone, iPad, or iPod touch to a new device ...