We have a duty days schedule, to perform support to our customers 7 days a week. One Saturday in a month - each of us has a duty day in couple with someone from the developers. Sunday is the customer-success-team lead duty day.
On duty we work in a usual weekday mode, answering dialogues and solving customers' tasks. But we should consider the fact, that we can't solve all dialogues and tasks immediately, being the only operator, who is online at the moment. That is why an answering time can be extended. We tell about this to our customers at the start of the conversation: We are online from 6:00 to 16:00 GMT on weekdays. Today we are not always online, so the response time may be extended..
If the issue is critical - the tracker doesn't work, or there's any other blocking issue that stops the tracker - we are trying to solve the customer's problem immediately. Ask for developer-on-duty's help. If you don't get the response in 1 hour, escalate to the team.
Feel free to methodically close dialogues and solve issues, without panicking that you are alone with a huge number of requests around.
If we face a question, which needed to be investigated with colleagues - tell the client that we will start solving it on Monday.
If there's an issue with the payment, extend the license to Friday next week, create a task and tell the customer we will solve everything on Monday.