  In this work, we study the exclusive decay of $\Upsilon$ into $J/\psi$ in
association with $\eta_c$ ($\chi_{c0,1,2}$). The decay widths for different
helicity configurations are evaluated up to QCD next-to-leading order within
the nonrelativistic QCD framework. We find that the QCD corrections notably
mitigate the renormalization scale dependence of the decay widths for all the
processes. The branching fraction of $\Upsilon\rightarrow J/\psi+\chi_{c1}$ is
obtained as $3.73^{+5.10+0.10}_{-2.06-1.19}\times 10^{-6}$, which agrees well
with the Belle measurement, i.e., ${\rm Br}(\Upsilon\rightarrow
J/\psi+\chi_{c1})=(3.90\pm1.21\pm0.23)\times10^{-6}$. For the other processes,
our results of the branching fractions are compatible with the upper limits
given by the Belle experiments, except for $\Upsilon(2S)\to J/\psi +\chi_{c1}$,
where some tension exists between theory and experiment. Having the polarized
decay widths, we study the $J/\psi$ polarization, which turn out to be
independent of any nonperturbative parameters. Further, according to our
calculation, it is promising to measure all the processes at Super B factory
thanks to the high luminosity.
