Learn English – Meaning of “January of next year” in December

When do you leave?
When are you leaving?

I would interpret this as a question about flight (or ferry/boat) times, or as a question about which day the flight would be on. I would expect a specific answer, such as the day or an approximate time on a day.

When do you leave?

Our flight is at 4 AM.
We leave on Friday.

When are you going?

I would expect this question only to be asked if the trip was not in the near future, and I would only expect a general answer – a season, month, or week.

When are you going?
(or: When do you go?)

The trip is in June.
We're going next spring.
We leave the second week of September.

Month names do not take the definite article except when you're concerned to distinguish a particular month from other months of the same name (e.g., the January when we had such warm weather--was that 1980 or 82?).

A day-of-month designator expressed as an ordinal number (first, second, third, &c) may take the definite article if it follows the month name, and must take the definite article if it is the head of the date expression:

So:

on January the first
on January first
on the first of January are all acceptable, but

on the January first
on the January the first
on first of January
on the first of the January
on first of the January are all unacceptable.