Addition and multiplication in terms of arrows in the poset category where an existing arrow means "divides".

There's no hope of recovering addition as a categorical construction in this poset, because addition isn't preserved under automorphisms of this poset. Write $\nu_p(n)$ for the largest exponent of a prime $p$ dividing $n$. We have

$$m \mid n \Leftrightarrow \nu_p(m) \le \nu_p(n)$$

which means that, as a poset, the divisibility poset is a (restricted) product of chains, one for each prime. In particular it has automorphisms given by permutations acting on the primes (which switch around the exponents in prime factorizations), and addition is totally scrambled by any such automorphism. But automorphisms preserve products, coproducts, and any other purely categorical constructions.

This objection doesn't apply to multiplication so one might hope for a categorical construction that produces multiplication but I am not seeing it. The problem basically reduces to the problem for a single chain: that is, in a chain $0 \le 1 \le 2 \le 3 \dots $ can you recover addition using only categorical constructions? I don't see a way to do this.

The most interesting thing people do with this poset that I know of is to prove Mobius inversion, which has a generalization to posets.

There's very little you can do in this poset. (I'm going to eschew the language of category theory, and just talk in terms of posets, since I think that makes things much clearer.)

Automorphisms of this poset are basically the same as permutations of the primes. Consequently, anything not invariant under permuting primes is not definable in any sense in that poset. So we can easily show for example that addition isn't definable: the automorphism $\alpha$ generated by the permutation of primes swapping $5$ and $7$ and leaving all other primes fixed does not respect addition (we have $2+3=5$ but $\alpha(2)+\alpha(3)\not=\alpha(5)$).

(A relevant topic you may be interested in is Skolem arithmetic; this is the first-order theory of the natural numbers with multiplication. This is actually more expressive than the first-order theory of the poset in question, but if we go a bit beyond first-order logic the two structures are appropriately equivalent.)