  We report $ab~initio$ calculations of the electronic and vibrational
properties in CuPb$_9$(PO$_4$)$_6$O, including the electron-phonon coupling
strength via strong-coupling Migdal-Eliashberg theory. We verify the presence
of appealing flat electronic bands near the Fermi level, a strong hybridization
between the Cu $3d$ and O $2p$ states, and soft low-energy phonons, which can
suggest high-temperature superconducting behavior. However, the electron-phonon
coupling strength appears insufficient to overcome the Coulomb repulsion
between an electron pair and thus does does not support high-temperature
superconductivity in CuPb$_9$(PO$_4$)$_6$O via the conventional electron-phonon
Migdal-Eliashberg mechanism. Even neglecting Coulomb repulsion of the electron
pair we find this electron-phonon coupling suggests a superconducting
transition temperature less than 2~K.
