  We propose a new way for sympathetic cooling of ions in an electromagnetic
Paul trap: it implies the use for this purpose of cold buffer atoms in the
region of atom-ion confinement-induced resonance (CIR). The problem is that the
unavoidable micromotion of the ion and the long-range nature of its interaction
with the environment of colder atoms in a hybrid atomic-ion trap prevent its
sympathetic cooling. We show that the destructive effect of ion micromotion on
its sympathetic cooling can however be suppressed in the vicinity of the
atom-ion CIR. The origin of this is the "fermionization" of the atom-ion wave
function near CIR, where the atom-ion pair behaves as a pair of noninteracting
identical fermions. This prevents the complete approach of the atom with the
ion near resonance and does not enhance the ion micromotion, which interferes
with its sympathetic cooling. We investigate the effect of sympathetic cooling
around CIRs in atom-ion and atom-atom confined collisions within the
qusiclassical-quantum approach using the Li-Yb$^+$ and Li-Yb confined systems
as an example. In this approach, the Schr\"odinger equation for a cold light
atom is integrated simultaneously with the classical Hamilton equations for a
hotter heavy ion or atom during collision. We have found the region near the
atom-ion CIR where the sympathetic cooling of the ion by cold atoms is possible
in a hybrid atom-ion trap. We also show that it is possible to improve the
efficiency of sympathetic cooling in atomic traps by using atomic CIRs.
