I am quite new in commutative algebra so I would be grateful if you could clarify to me this issue.
I would like to construct the coordinate ring, $A$, of a Calabi Yau threefold $X$ which is a complete intersection of two hypersurfaces $h_{1}$ and $h_{2}$ in the ambient space $\mathcal{A}=\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{2}$.
We can choose projective coordinates: $x=[x_{0}:x_{1}] $ for $\mathbb{P}^{1}$, $y=[y_{0}:y_{1}:y_{2}]$ for the first $\mathbb{P}^{2}$ and $z=[z_{0}:z_{1}:z_{2}]$ for the other $\mathbb{P}^{2}$. The configuration matrix is: (it is the Schoen manifold)
\begin{equation} X\left[\begin{array}{c||ccc} \mathbb{P}^{1}&1& 1\\ \mathbb{P}^{2}&0& 3\\ \mathbb{P}^{2}&3& 0\\ \end{array}\right]= \begin{cases} x_{0}f_{0}(y)-x_{1}f_{1}(y)=0& \\ x_{0}g_{0}(z)-x_{1}g_{1}(z)=0 & \end{cases} \end{equation} where $f_{0}, f_{1}, g_{0}, g_{1}$ are homogeoneous cubic polynomials.
I know that for a single projective factor, for example the quintic in $\mathbb{P}^{4}$ I can use $A=R_{\mathbb{P}^{4}}/I=\mathbb{C}[x_{0},x_{1},x_{2},x_{3},x_{4}]/\left ( \sum_{i=0}^{4} x_{i}^{5}+\psi x_{0}x_{1}x_{2}x_{3}x_{4} \right )$
EDIT1 My initial guess was that: \begin{gather} \nonumber A =(\mathbb{C}[x_{0},x_{1},y_{0},y_{1},y_{2},z_{0},z_{1},z_{2}]/ \left( x_{0}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{a} y_{0}y_{1}y_{2})-x_{1}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{b} y_{0}y_{1}y_{2}) \\ , x_{0}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{c} z_{0}z_{1}z_{2})-x_{1}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{d} z_{0}z_{1}z_{2}) \right ) \end{gather}
EDIT2 I think I will follow equation $(3.3)$ of this in which the coordinate ring of a CICY is explained.
This isn't correct. The homogenous coordinate ring doesn't behave this way with respect to taking products, essentially since $\mathbb{P^n}\times \mathbb{P^m}$ is not $\mathbb{P^{n+m}}$. See this answer for some explicit examples of coordinate rings of products, Homogeneous coordinate rings of product of two projective varieties.
What you'd have to do here is embed $\mathcal{A}$ into a projective space $\mathbb{P^N}$ via the Segre embedding, write the equations defining $X$ in the coordinates on $\mathbb{P^N}$, then the homogenous coordinate ring will be the polynomial ring in $N+1$ variables quotiented by the polynomials defining $\mathcal{A}$, and the (new) polynomials defining $X$.