Current protocol of Amide Proton Transfer-weighted (APTw) imaging commonly starts with the acquisition of high-resolution T2-weighted (T2w) images followed by APTw imaging at particular geometry and locations (i.e. slice) determined by the acquired T2w images. Although many advanced MRI reconstruction methods have been proposed to accelerate MRI, existing methods for APTw MRI lacks the capability of taking advantage of structural information in the acquired T2w images for reconstruction. In this paper, we present a novel APTw image reconstruction framework that can accelerate APTw imaging by reconstructing APTw images directly from highly undersampled k-space data and corresponding T2w image at the same location. The proposed framework starts with a novel sparse representation-based slice matching algorithm that aims to find the matched T2w slice given only the undersampled APTw image. A Recurrent Feature Sharing Reconstruction network (RFS-Rec) is designed to utilize intermediate features extracted from the matched T2w image by a Convolutional Recurrent Neural Network (CRNN), so that the missing structural information can be incorporated into the undersampled APT raw image thus effectively improving the image quality of the reconstructed APTw image. We evaluate the proposed method on two real datasets consisting of brain data from rats and humans. Extensive experiments demonstrate that the proposed RFS-Rec approach can outperform the state-of-the-art methods.