Azure AD Connect is typically a one-way sync, so there's no additional data written back to the AD Object.
Our AD Connect runs every 30 minutes and in general, as long as the account is in an OU (or group if you have it set that way) that is configured to sync, it goes up without an issue. I get emailed via the Technical Contact address for any DirSync errors that would prevent a synchronization.
You can get a lot of info about a user via the Azure PowerShell Module.
https://technet.microsoft.com/en-us/library/dn975125.aspx?f=255&MSPPError=-2147217396
I don't know if there's a way to just do a remote session through regular PowerShell, but it's a small install to get the Azure PS.