To avoid this problem, set the go-live date after the next sync. For example, if you have a daily sync, schedule it for the next day. Web alerts do not fire in a staging environment by design -- web alerts are only designed to notify when content goes live on a production machine.
Important: For web alerts to work as designed, the staging environment must have the server type set to Enable Staging Server, and production should be the opposite. Set this value in the setup screen (Workarea Settings Configuration Setup).