Box dimension of the graph of an increasing function
Pietro Majer's argument that you cited actually shows that the upper box dimension is $1,$ and hence the lower box, the upper and lower packing, and the Hausdorff dimensions are all equal to $1.$ Also, graphs of real-valued Lipschitz functions defined on an interval (of positive length) have upper box dimension $1,$ and as Pietro Majer also pointed out there in a comment, the graph of a strictly increasing continuous function is geometrically congruent to the graph of a Lipschitz function.
Regarding graphs of Lipschitz functions, see Corollary 11.2(a) [using $s=1$] on p. 147 of Falconer's 1990 book Fractal Geometry. Regarding graphs of monotone and bounded variation functions, see p. 120 (beginning of Section 10.4) and Section 12.4 (pp. 148-150) of Tricot's 1995 book Curves and Fractal Dimension and this paper.
Let me encode the solution of the problem explicitly. From the linked answer we had:
Theorem If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$\mathcal{H}^1(\Gamma)=L(\gamma).$$
This theorem applies in our context since we have a strictly increasing function and:
Theorem Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.
Our curve is rectifiable, that is is length is finite, choosing the sum distance:
$ L(\gamma) = \sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\} = \sup\left\{\sum_{i=1}^{n-1}(t_{i+1}-t_i) + (\gamma(t_{i+1})-\gamma(t_i)) \right\} = 2 $
So the theorem applies and we get $0 < \mathcal{H}^1 < \infty$ which implies $\dim_H(graph(f)) = 1$.
As it was pointed out in the accepted answer, Falconer's Fractal Geometry contains in Corollary 11.2 a result which can be adapted in our context to:
Let $f:[0,1] \to \mathbb{R}$ a Lipschitz function, then $\dim_H(graph(f)) \le \dim_B(graph(f)) \le 1$
So in conclusion we get $dim_H(graph(f)) = dim_B(graph(f)) = 1$. In summary we have obtained that:
If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $dim_H(graph(f)) = dim_B(graph(f)) = 1$.