The OP seems to have a pretty good handle on the question and maybe just needs to think it through to be fully satisfied.
But here is a slightly different approach which makes it even more clear (to me) that the OP's method will work. This was borne out of the remark I made (to myself) upon reading the OP's clarification that the natural numbers include zero (you're damn right they do, by the way): namely, it certainly doesn't matter, because any two countably infinite discrete spaces are homeomorphic.
Taking that one step further, it is clearly enough to realize $\mathbb{Z}$ with the discrete topology as a continuous image of the Sorgenfrey line: having done this, compose with any homeomorphism (i.e., bijection!) from $\mathbb{Z}$ to $\mathbb{N}$. (Or, in fact, with any surjection, as the OP has done.) For this, literally take the greatest integer function. The preimage on any given basis element -- i.e., a singleton set $\{n\}$ -- is the half-open interval $[n,n+1)$. I don't keep too much information about the Sorgenfrey line in my head, but I'm pretty sure those sets are open!