We present the results of Suzaku X-ray follow-up observation of weak-lensing-detected halos in the field around galaxy cluster ZwCl0823.2+0425. We clearly detected X-ray emission associated with most of these halos and determined their detailed physical parameters such as X-ray luminosity, temperature, and metal abundance, for the first time. We find that the X-ray luminosity - temperature relation for these halos agrees with former typical results. With mass determined from the weak gravitational lensing data, the mass-temperature relation for them is also investigated and found to be consistent with the prediction from a simple self-similar model and results of the previous studies with both lensing and X-ray data. We would like to emphasize that the self-similar scaling relation of mass and temperature is shown here for the first time using a weak-lensing selected sample, whereas previous studies of the mass scaling relation used X-ray-selected samples of clusters. Therefore, our study demonstrates importance of X-ray follow-up observations of shear-selected clusters, and shows that a joint X-ray and lensing analysis will be crucial for clusters discovered by the forthcoming weak-lensing surveys, such as the one planned with Subaru/Hyper-Suprime-Cam.