My understanding is that alchemy was a precursor to modern chemistry. Some might say that numerology, similarly, was an earlier form of what is now known as number theory, but I feel like it's a more complicated proposition.
At it's heart, numerology is about playing games with and finding relationships between numbers, which seems very similar to number theory. The key distinction may be that number theory is scientific where numerology is mystical, with perhaps some overlap prior to the Enlightenment, Pythagoreanism being an example.
My feeling is that arithmetic is very old indeed, and numerology cannot be said to precede arithmetic because it requires arithmetic.
But I also know there is a distinction between classical arithmetic and modern number theory, so I'd welcome any additional insight, correction, or related information in support or against this idea.
Note: The impetus for this question came out of a recent question on Stack Mythology regarding the origin of 12 Olympians specifically. In a mythological context such as this, numerological ideas can hold some validity.
Although contemporary numerology is largely concerned with non-scientific prediction and non-scientific attempts to influence reality, I have also read convincing arguments that it may have been used as method of encoding esoteric knowledge, indicating a possible, practical use.
The most satisfactory answer to the question in question was related to the practical aspects of the integer 12.