Constructing a Turing machine which decides whether a fixed TM will halt on a fixed input or not
Since $M_0$ and $w_0$ are fixed parameters of the problem, the answer is yes: for every fixed $M_0$ and $w_0$, there exists a Turing machine $M_{M_0, w_0}$ (depending on  $M_0$ and $w_0$) such that, for the input $(M_0, w_0)$, $M_{M_0, w_0}$ returns $1$ if $M_0(w_0)$ halts and 0 otherwise.

In particular one such Turing machine $M_{M_0, w_0}$ must be one of the following two machines:


$M'_1$: Write $1$. Halt.
$M'_0$: Write $0$. Halt.


If, instead, you are looking for an algorithm that takes $M_0$ and $w_0$ as input and outputs a machine $M_{M_0, w_0}$ with the above property, then you are out of luck: there is not such algorithm in general (it might exist if you restrict the set of input machines $M_0$).
Suppose that such an algorithm (i.e., Turing machine) $A$ existed, then it would allow to solve the halting problem:


Given $M_0$ and $w_0$, compute $M_{M_0, w_0}$ by simulating $A$ on input $(M_0, w_0)$.
Simulate $M_{M_0, w_0}$ on input $(M_0, w_0)$. By definition of $M_{M_0, w_0}$ this step requires finite time.
Return "yes" if the output of $M_{M_0, w_0}( (M_0, w_0) )$ was $1$, otherwise return "no".