= Solution
Introduce $Z_{ij}\in\{0,1\}$, where $Z_{ij}=1$ means that the observation came from the structural-zero component. Conditional on the parameters,
$$
\Pr(Z_{ij}=1\mid Y_{ij})=
\begin{cases}
\displaystyle\frac{\pi_j}{\pi_j+(1-\pi_j)e^{-\alpha_i\beta_j}},&Y_{ij}=0,\\
0,&Y_{ij}>0.
\end{cases}
$$
Given $Z$, the nonstructural observations are independent Poisson variables. Using shape-rate parameterization and the stated unit-rate priors, the remaining Gibbs updates are
$$
\alpha_i\mid-\sim\operatorname{Gamma}\left(
1+\sum_j(1-Z_{ij})Y_{ij},
1+\sum_j(1-Z_{ij})\beta_j
\right),
$$
$$
\beta_j\mid-\sim\operatorname{Gamma}\left(
1+\sum_i(1-Z_{ij})Y_{ij},
1+\sum_i(1-Z_{ij})\alpha_i
\right),
$$
$$
\pi_j\mid-\sim\operatorname{Beta}\left(
1+\sum_iZ_{ij},
1+n-\sum_iZ_{ij}
\right).
$$
Alternating these four standard-distribution updates defines the requested <Gibbs sampler> for the <Zero-inflated Poisson distribution> posterior.
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