The honeycomb-lattice Ising antiferromagnet subjected to the imaginary magnetic field $H=itheta T /2$ with the topological angle $theta$ and temperature $T$ was investigated numerically. In order to treat such a complex-valued statistical weight, we employed the transfer-matrix method. As a probe to detect the order-disorder phase transition, we resort to an extended version of the fidelity $F$, which makes sense even for such a non-hermitian transfer matrix. As a preliminary survey, for an intermediate value of $theta$, we investigated the phase transition via the fidelity susceptibility $chi_F^{(theta)}$. The fidelity susceptibility $chi_F^{(theta)}$ exhibits a notable signature for the criticality as compared to the ordinary quantifiers such as the magnetic susceptibility. Thereby, we analyze the end-point singularity of the order-disorder phase boundary at $theta=pi$. We cast the $chi_F^{(theta)}$ data into the crossover-scaling formula with $delta theta = pi-theta$ scaled carefully. Our result for the crossover exponent $phi$ seems to differ from the mean-field and square-lattice values, suggesting that the lattice structure renders subtle influences as to the multi-criticality at $theta=pi$.
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