The work is done in the two weeks before you go, not during. Almost everything that makes this survivable is preparation.
Pull your last two hundred tickets and count the categories. You will find a small number of questions account for most of the volume, usually password and login, billing and invoices, one confusing part of the product, and one bug people keep hitting. That list is your whole task.
Then, for each:
Write the answer once, properly, somewhere public. A help page, a short doc. Not because people read documentation, but because you can link to it in one line instead of typing the answer.
Fix the one that is a product problem. If a fifth of your tickets are people confused by the same screen, two days of work before your holiday removes a fifth of your inbox permanently. This is the highest-return thing in this entire thread and people never do it, because answering feels urgent and fixing feels optional.
Put the top three answers where the question is asked - next to the billing page, next to the confusing screen, in the signup email.
Do that and the fortnight's volume is meaningfully lower before you leave, which changes every other option from impossible to manageable.