My attempt:
Let $\epsilon$>0, as $f\in R_{\alpha}[a,b]$ there is a partition $P_{\epsilon}$ of $[a,b]$ such that $U(P_{\epsilon},f,\alpha)-L(P_{\epsilon},f,\alpha)<\epsilon$. To prove the continuity of $f$ in $c\in[a,b]$ I can take $\delta=||P_{\epsilon}||$, then for every $x$ with $|x-c|<\delta$ taking $P=P_{\epsilon}\cup\{x,c\}$, we have:
$$U(P,f,\alpha)-L(P,f,\alpha)<\epsilon$$
But $U(P,f,\alpha)-L(P,f,\alpha)=H+(M_{i}-m_{i})(\alpha(x)-\alpha(c))$, where H is the sum over the other intervals. And $|f(x)-f(c)|\leq M_{i}-m_{i}$, hence:
$$|f(x)-f(c)|(\alpha(x)-\alpha(c))<\epsilon$$
The conclusion is only possible if the function $\alpha$ is strictly increasing; otherwise, I cannot guarantee anything, so I think there might be some counterexample.
Commentary:
When we take $P=\{x_{0},x_{1},...,x_{n}\}$ a partition of the interval $[a,b]$ with $a=x_{0}<x_{1}<...<x_{n-1}<x_{n}=b$, we define:
$U(P,f,\alpha)= \sum_{i=1}^{n}M_{i}\Delta\alpha_{i}~~$ and $~~L(P,f,\alpha)= \sum_{i=1}^{n}m_{i}\Delta\alpha_{i}$
with $\Delta\alpha_{i}=\alpha(x_{i})-\alpha(x_{i-1})$, $M_{i}=sup\{f(x): x\in[x_{i-1},x_{i}]\}$ and $m_{i}=inf\{f(x): x\in[x_{i-1},x_{i}]\}$.
$\alpha$ denotes any monotone increasing function: $\alpha:[a,b]\to\mathbb{R}$