Recently, the performance of monocular depth estimation (MDE) has been significantly boosted with the integration of transformer models. However, the transformer models are usually computationally-expensive, and their effectiveness in light-weight models are limited compared to convolutions. This limitation hinders their deployment on resource-limited devices. In this paper, we propose a cross-architecture knowledge distillation method for MDE, dubbed DisDepth, to enhance efficient CNN models with the supervision of state-of-the-art transformer models. Concretely, we first build a simple framework of convolution-based MDE, which is then enhanced with a novel local-global convolution module to capture both local and global information in the image. To effectively distill valuable information from the transformer teacher and bridge the gap between convolution and transformer features, we introduce a method to acclimate the teacher with a ghost decoder. The ghost decoder is a copy of the student's decoder, and adapting the teacher with the ghost decoder aligns the features to be student-friendly while preserving their original performance. Furthermore, we propose an attentive knowledge distillation loss that adaptively identifies features valuable for depth estimation. This loss guides the student to focus more on attentive regions, improving its performance. Extensive experiments on KITTI and NYU Depth V2 datasets demonstrate the effectiveness of DisDepth. Our method achieves significant improvements on various efficient backbones, showcasing its potential for efficient monocular depth estimation.