Back-up iPhone contents

Try How to back up your iPhone, iPad, and iPod touch - Apple Support

Backup is all or nothing deal, you cannot select what you want to backup.

Note: If you have Settings > Apple ID > iCloud > Contacts ON, it will automatically saved on iCloud.

You can access it using a browser https://www.icloud.com/#contacts

or you can sign up on a new device with the same Apple ID and password and it auto populate the contacts on that device.