The length of these curves is unbounded. For a positive integer $n$ consider a triangular wave $f_n:[0,1]\to\mathbb{R}$ with support on $[1/3,2/3]$, making $n$ (isosceles) triangular impulses on $[1/3,2/3]$ with $\|f_n(x)\|_\infty=\frac{1}{5\sqrt n}$. The graph of $f_n$ is a curve satisfying the monotonicity constraint, with length larger than $2\sqrt n/5$.

The length of these curves is unbounded. For a positive integer $n$ consider a triangular wave $f_n:[0,1]\to\mathbb{R}$ with support on $[1/3,2/3]$, making $n$ (isosceles) triangular impulses on $[1/3,2/3]$ with $\|f_n(x)\|_\infty=\frac{1}{6\sqrt n}$. The graph of $f_n$ is a curve satisfying the monotonicity constraint, with length larger than $\sqrt n/3$.