How to recover old contacts in iCloud?

If you were syncing Contacts with iCloud, go to iCloud.com using a computer and select Account Settings. When that page loads, look at the bottom left side under Advanced and see if you can restore from there. 

How to Recover Files, Contacts, Calendars and Reminders in iCloud

If you weren't syncing Contacts with iCloud, do you have a backup you can restore from? Restoring from a backup erases all data currently on your device. Any data newer than when the backup was made will be erased. A restore is an all or nothing restore. You can't access a backup to see the contents and only restore certain files.

Backup - Restore your device from an iCloud or iTunes backup.