We have developed multi-channel apparatus for electron-atom Compton scattering to study the momentum distribution of atoms in a molecule. It combines the features of both a spherical electron energy analyzer and a large-area position sensitive detector, thereby having an ability to cover almost completely the azimuthal angle range available for quasi-elastic electron Rutherford backscattering at an angle of 135°. Details and performance of the apparatus are reported, together with experimental results measured for Xe and CH4 at an incident electron energy of 2 keV. In particular, it is shown that the instrumental sensitivity is remarkably high, which has increased the signal count rate by nearly three orders of magnitude compared to existing setups. This technical progress would be useful for advancing atomic momentum spectroscopy studies.