Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional derivative $$ D^\alpha f(x)=C_\alpha \frac{d}{dx}\int_0^x(x-y)^{-\alpha}f(y)\,dy,\quad \alpha\in(0,1), $$ where $f\in \Lambda([0,1])$, $f(0)=0$? Would it be $C^{1-\alpha}([0,1])$?

It is indeed what you expect. A rather straightforward proof of this fact when $[0,1]$ is replaced with $\mathbb{R}^n$ relies on Littlewood-Paley decomposition and is detailed for instance in this book. Since your definition of $\Lambda$ is only local, you may simply extend your functions to $\mathbb{R}$ by multiplying them with a smooth cutoff and then use the results from the book.