After doing authentication (using prometheus basic authentication), how long authentication takes ? and how to customize authentication expiration?
It seems like you misunderstood something.
Basic authentication doesn't have any expiration, because your authentication data have to be transferred with every request.
You haven't provided much information on your problem, but I will assume based on the wording of a question that you are talking about authentication at Prometheus' web-interface using browser.
If this is the case then it's something about caching credentials by browser.
You can't configure this at Prometheus' side.
AFAICT, this is not configurable in browsers too, but you could ask it at superuser (as it's not really on topic here)