Hi Customer,
In DirSync, if you have password sync feature enabled (tick the password sync option when running the DirSync configuration wizard), then the on-premises passwords will be synced to Office 365 and the user can use the on-premises password to sign into Office 365. If the password sync feature is not enabled, then the user still needs to use the Office 365 password to sign in. please note that the Password Sync feature of the Directory Sync tool will not work correctly if Directory Sync tool is deployed on an OS older than Windows Server 2008 R2 SP1.
Also, DirSync is not supported to be installed on a Small Business Server, you will need to set up a new server and deploy the DirSync tool on the server. For more details, please refer to: Prepare for directory synchronization.