Welcome to Tableau Online!
To answer your questions:
1. Depending if you have a live or extracted data sources, the alert will be checked as follows:
Note: For live data sources, alerts are checked every 60 minutes for Tableau Online, or on a frequency set by administrators for Tableau Server. Extracts are checked each time they're refreshed.
Based on how the data alerts are checked and that you have chosen 'Once - the first time it's true', I believe this is a true 'one time only' as you mentioned. Depending on your use case you may be better using the other options.
Another thing to note, if the alert has evaluated to true, then Tableau Online will not check again until the given email interval has passed (Hourly at most, Daily at most, etc). One exception is that if you edit the alert & save changes, the clock is reset. In this case, the alert will be evaluated upon save and we will notify you again if the new condition evaluates to true. In this case, it’s essentially treated like it’s a new alert.
2. You should not have to recreate the alert if the workbook is overwritten with the same name. If the workbook is published with a different name then yes. All your current alerts can be found under 'Alerts' in the 'My Account Settings' section of Tableau Online.
I hope this helps!
Awesome, thanks Jason!