  A nuclear-spin exchange interaction exists between two ultracold fermionic
alkali-earth (like) atoms in the electronic $^{1}{\rm S}_{0}$ state ($g$-state)
and $^{3}{\rm P}_{0}$ state ($e$-state), and is an essential ingredient for the
quantum simulation of Kondo effect. We study the control of this spin-exchange
interaction for two atoms simultaneously confined in a quasi-one-dimensional
(quasi-1D) tube, where the $g$-atom is freely moving in the axial direction
while the $e$-atom is further localized by an additional axial trap and behaves
as a quasi-zero-dimensional (quasi-0D) impurity. In this system, the two atoms
experience effective-1D spin-exchange interactions in both even and odd partial
wave channels, whose intensities can be controlled by the characteristic
lengths of the confinements via the confinement-induced-resonances (CIRs). In
current work, we go beyond that pure-1D approximation. We model the transverse
and axial confinements by harmonic traps with finite characteristic lengths
$a_\perp$ and $a_z$, respectively, and exactly solve the "quasi-1D + quasi-0D"
scattering problem between these two atoms. Using the solutions we derive the
effective 1D spin-exchange interaction and investigate the locations and widths
of the even/odd wave CIRs for our system.
  It is found that when the ratio $a_z/a_\perp$ is larger, the CIRs can be
induced by weaker confinements, which are easier to be realized experimentally.
  The comparison between our results and the recent experiment shows that the
two experimentally observed resonance branches of the spin-exchange effect are
due to an even-wave CIR and an odd-wave CIR, respectively. Our results are
advantageous for the control and description of either the effective
spin-exchange interaction or other types of interactions between ultracold
atoms in quasi 1+0 dimensional systems.
