Geometrically, this asks if every Noetherian connected affine scheme can be realized as a closed subscheme of a Noetherian integral affine scheme.
To briefly justify this question: the linked answer above shows that the approach to the original question is doomed to fail. Moreover, I would argue that this question is inherently more natural, in the following sense: given Noetherian rings $R_1, \ldots R_n$, there is a Noetherian domain $D$ surjecting onto each of $R_1, \ldots, R_n$ iff there exists a Noetherian domain $D'$ surjecting onto $R_1 \times \ldots \times R_n$. Thus the original question is isomorphic to a direct product of copies of this one.
Since I expect answers will reduce to this case anyway, let me make a mild regularity assumption and only consider reduced (i.e. $R_0 + S_1$) rings. Note however that even a slight improvement on this, like normality ($R_1 + S_2$), would render the problem trivial.