We study the first-passage-time (FPT) properties of active Brownian particles to reach an absorbing wall in two dimensions. Employing a perturbation approach, we obtain exact analytical predictions for the survival and FPT distributions for small Péclet numbers, measuring the importance of self-propulsion relative to diffusion. While randomly oriented active agents reach the wall faster than their passive counterpart, their initial orientation plays a crucial role in the FPT statistics. Using the median as a metric, we quantify this anisotropy and find that it becomes more pronounced at distances where persistent active motion starts to dominate diffusion.