How to solve this mapping reduction problem by proving the computability of this function?
The reduction function is simply

$$
f(x)=x0
$$

which is computable, since a TM can move to the end of $x$, append a $0$, and halt. It is also total, by definition.

Note that to prove that $f$ is computable we do NOT have to consider the decidability / enumerability of any sets. I think your confusion is caused by thinking about c.e. sets when crafting $f$, but defining a reduction function does not involve enumerability. You only have to ensure that $f$ is total computable, and that $x\in A \iff f(x)\in B$ for all $x$.