What is an estimation of the Kolmogorov Complexity for the first N integers?
No, there is no general algorithm to compute a close approximation to the Kolmogorov complexity of the sequence $1,2,\dots,n$.  Any candidate algorithm you come up with will have some inputs where it gives a bad answer (a poor approximation to the correct answer).

Denote by $[n]$ the binary encoding of the natural number $n$, and let $[[n]] = [1],[2],\ldots,[n]$ denote a comma-separated encoding of the sequence $1,\ldots,n$. It is not hard to check that $|K([n]) - K([[n]])| = O(1)$.  Since computing $K([n])$ is undecidable, it follows that computing a good approximation to $K([[n]])$ is also undecidable.

Furthermore, it is known that for "most" integers $n$, $K([n]) = \Theta(\log n)$. So for "most" integers $n$, $K([[n]]) = \Theta(\log n)$.