  We perform simulation study for a 10-kg $^{100}$Mo-based bolometeric
demonstration experiment for neutrinoless double-beta decay ($0\nu\beta\beta$)
search at China JinPing underground Laboratory (CJPL). Cosmogenic production of
radionuclides in $^{100}$Mo-enriched lithium molybdate crystals and copper
components of the detector system are studied using Geant4 toolkit based on the
simulated cosmic ray data from the CRY library. Background energy spectra of
the cosmogenic radionuclides including $^{56}$Co, $^{82}$Rb and $^{88}$Y which
are harmful for the $^{100}$Mo-based $0\nu\beta\beta$ experiment are
investigated. We then evaluate the total cosmogenic background level in the
$^{100}$Mo $0\nu\beta\beta$ search energy region of interest (ROI) for the
demonstration experiment. After one year of cooling down underground, the
residual background contribution is found to be 1.8$\times$10$^{-6}$
cts/kg/keV/yr and 3.3$\times$10$^{-4}$ cts/kg/keV/yr from crystals and copper
components, respectively. Furthermore, underground cosmogenic activation of
copper and lithium molybdate crystal is calculated based on the simulation
spectra of neutron and proton in CJPL. The underground cosmogenic background is
found to be negligible in the ROI.
