I manage quite a few Office 365 setups each using Azure connect synchronizing local AD to Office 365. When something is wrong with synchronizations I get alerts which is good however if password synchronization stops working then no alerts occur. This morning I had a client where password syncs were not working for over three weeks and I had no idea until logged onto Office 365 and noticed 'No recent synchronizations' next to the 'Password sync status'. Last 'Last directory sync' was however working and showing up to date.
I fixed this by going through the setup wizard for Azure connect again and all started working however I would like to be alerted if there is a 'No recent password synchronization error'.