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The category Poset has posets as objects and monotonic functions as arrows.

My understanding is that as long as the composite of arrows is associative (and all objects are indeed posets), then it is still considered a category.

In that case, the arrows can simply be a total function, and this version of Poset would still be a category. Because even if a function swaps the ordering of elements inside the poset, the resulting poset is still a poset.

Is my understanding correct or am I missing something?

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Yes, it is correct. But it does not produce a new category: your category of posets and non-monotonic functions is equivalent to $\mathbf{Set}$.

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