Multispectral endomicroscopy provides tissue functional information in addition to structural information for accurate disease diagnosis. In this Letter, we propose a snapshot multispectral endomicroscope that employs a fiber bundle to deliver an in-body tissue spatial-spectral datastream to an external compressive spectral imager. Equipped with an end-to-end deep-learning-based reconstruction algorithm, we are able to capture tissue multispectral data in video rates and reconstruct high-resolution multispectral images with up to 24 spectral channels in near-real time.