Such a number isn't really that big. You could do long division by hand and, even if you assume that it takes 10 seconds to extract each digit (which is likely an overestimate considering that you will gain substantial practice dividing by this specific divisor of $23$ by the time you are done), you're still only talking about 50 hours of work - not at all unreasonable and possible since antiquity. There used to be teams who did this as a job.
If you wanted to check such a large number only for divisibility by $23$, likely the easiest method would be to observe that
$$10^{22} \equiv 1 \pmod{23}$$
meaning that to check if a number is divisible by $23$, you can just break it (going from right to left) into blocks of $22$ digits, sum up all the blocks, and see if the result is a multiple of $23$ - that'd be pretty boring as well and possibly less accessible to people from so long ago (though it's hard to say when, exactly, such a method became common knowledge - certainly by the middle of the 17th century at latest) - but you'd only have to do $824$ addition of numbers up to $25$ digits, followed by an easy division of a $25$ digit number by $23$. You could also easily have multiply people work at such a method - with enough help, you could surely get this done in an hour and you'd then know the remainder of your number divided by $23$. Even working alone, doing single digit additions at just one per five seconds, you're talking about less than 30 hours of work this way (and I bet if you were focussed, you could do far more than one addition per five seconds).
You would have to ask a math historian about exactly when such a method might have been available - modular arithmetic (or, perhaps more in line with how it was imagined historically: the study of linear Diophantine equations) is very old and it is easy to underestimate how clever ancient mathematicians were with algebra. It's also worth remembering how different their views were to ours: if you're talking about things more than a thousand years ago, you can't even take base 10 positional notation for granted so the idea to concatenate the decimal digits of something might be very unnatural to people of the age.
In the modern age, we have computers. They can do long division much faster (or use other methods). For instance, I asked Mathematica to divide a $2000$ digit number by $23$. It took 11 microseconds. Even if you don't have fast computers, long division can be implemented (essentially optimally!) by a finite state machine with $23$ states and an alphabet of the $10$ digits - so even the most rudimentary computer would have no trouble doing this division.
That said, numerology like this has a problem: there are lots of extraordinary coincidences that could occur in a set of numbers. While each one may have probabilities of only a few in a thousand, if you look hard enough and examine many possibilities, you are likely to find something interesting - but you would find the same things in random numbers.
This sort of observation is similar to what would happen if you ran a lottery among $1000$ people and then started interrogating each person about whether they had won. Most would say no, and you would not publicize this - but one would say yes and you could rightly proclaim that it was extraordinary that so-and-so won because they only had a 1/1000 chance of doing so - and even though you did not know who you would find coming into this experiment, it was certain that you would find someone. This is the same as if you test a set of numbers by many criteria (not to mention that there are lots of old books and lots of statistics you could associate to each one) - you do not know the result you will find, but you will probably find something. This doesn't make that something meaningful though.
