Hi,
Background task is running every hour, so you no need to click on button manually.
You can update the Last Sync of Check in also automatically via below server script:
s = frappe.get_doc('Shift Type','Morning')
s.last_sync_of_checkin = frappe.utils.add_to_date(frappe.utils.today(), days=-1, as_string=True) + ' 23:59:59'
s.save()
Create server script with above code and select Script Type - Scheduler Event and Event Frequency- Daily.
This script will run at midnight and update the last sync for yesterday and time you can modify as per your requirement.
Thanks,