Is there an option to set the local backup to run upon device starting in the event that it was off at the designated backup time or it was disconnected from local backup?

The backup job will run at the scheduled time and in the event of error, the scheduler will re-attempt to backup based on policy configured in Error Handling option. The default policy is to attempt 300 times, with a 30 seconds interval.

For more details on Error Handling - http://www.acronis.com/en-us/support/documentation/Acronis_Backup_Cloud/#36342.html