Learn Hardcore Arabic (Modern Standard): الحجز في الفندق هذا الأسبوع، وليس في الشهر القادم. - The reservation at the hotel is this week, not next month.

In Arabic, the verb to be is usually omitted in the present tense. So a sentence like الحجز في الفندق هذا الأسبوع can naturally mean The reservation is at/in the hotel this week even though there is no separate word for is.

This is called a nominal sentence:

الحجز = the topic / subject
the rest = information about it

So English needs is, but Arabic usually does not.

Why does the second part use وليس instead of لا?

ليس is commonly used to negate nominal sentences or an implied present-tense is.

So:

وليس في الشهر القادم means and it is not in the coming month

Here:

و = and
ليس = is not

Using لا here would be less natural, because لا is more commonly used with verbs or other types of negation. Since the sentence is built like a present-tense X is Y statement, ليس is the expected choice.

Why doesn’t Arabic repeat الحجز in the second half?

Because it is already understood from the first half.

Arabic often leaves out repeated words when the meaning is clear: