Like you, I wasn't able to find that exact information. However, it appears that the recording method used in India at the time was palm-leaf manuscripts, and the oldest of those known to exist are from the 1st Century CE. Considering all those were found in areas ideal for preservation, that's probably a rough upper-bound on the physical age you could expect out of the manuscripts themselves.
That web page reports the oldest surviving palm-leaf manuscript in Sanscrit is from the 9th Century. Since Arthashastra is in Sanscrit, then assuming all of those copies have been dated, they wouldn't be older than that.
However, the age of the copy itself is not generally useful information for a layman. Modern people are spoiled by their cheap, perfect modern copying devices. For ancient people, copying was a necessity, because originals deteriorate (or suffer mishaps), but it was very expensive and every copy was imperfect.
This has a lot of repercussions that we spoiled 21'st Century humans aren't used to having to worry about. Here are some things modern people often don't quite grok about ancient manuscripts.
They are usually incomplete
The best oldest sources are (by virtue of being the oldest) in the worst states of repair when found. That means typically large parts tend to be missing or illegible.
They change over time
Every copy messes something up. Sometimes those will get "fixed" in later copies, but often not. Things that are surprising to the guy making the copy are more likely to get fixed, but that means surprising portions are likely to get slowly not fixed, or even "fixed", to be not so surprising.
They are inconsistent
This follows from the point above. Since every copy is slightly different, over the generations these differences will accumulate, and can actually lead to different copies having noticeable lineages, if you have enough of them to work it out.
Material gets added
"Copyright" didn't exist, so authorship was not a big deal. If you had some material you wanted spread, the absolute best way to do it was to attribute it to someone famous, and slip it into a copy of their work.
We can tell this happened today due to the fact that an author's vocabulary is like a fingerprint, and sections that have different "fingerprints" were clearly from different authors. You can even date vocabularies and themes to a certain extent. This is how we know that somewhere between 4 and 6 of Paul's Epistles were not written by Paul, and Arthashastra had at least 2 different authors (and likely many more).