Forecast a range, not a number
Look at the ticket volume after your last release or campaign and scale it for the size of this one. Plan for a low, expected and high case so the high case does not surprise you.
Write the replies before launch day
Draft saved replies and help articles for the questions you can already predict: pricing, how to switch, what changed. Have someone outside the launch team read them for clarity.
Plan the staffing for the first days
Schedule extra coverage for the first two or three days and keep the rest of the team free of other deadlines. Decide in advance who covers if volume is higher than the high case.
Set up a way to spot new problems
Create a launch tag and ask everyone to apply it. If a new issue appears, a lead should be able to see it within the hour, not at the end of the week.
Tell customers what to expect
If you will be slower than usual, say so in your auto-reply and on the contact page. People forgive a stated delay far more readily than silence.
Review after the first week
Look at volume, response times and the top issues, and update your articles and replies. Write down what you would change before the next launch while it is fresh.