Hyperfine structure (HFS) of atomic energy levels arises due to interactions of atomic electrons with a hierarchy of nuclear multipole moments, including magnetic dipole, electric quadrupole and higher rank moments. Recently, a determination of the magnetic octupole moment of the $^{173}\mathrm{Yb}$ nucleus was reported from HFS measurements in neutral ${}^{173}\mathrm{Yb}$ [PRA 87, 012512 (2013)], and is four orders of magnitude larger than the nuclear theory prediction. Considering this substantial discrepancy between the spectroscopically extracted value and nuclear theory, here we propose to use an alternative system to resolve this tension, a singly charged ion of the same $^{173}\mathrm{Yb}$ isotope. Utilizing the substantial suite of tools developed around $\mathrm{Yb}^+$ for quantum information applications, we propose to extract nuclear octupole and hexadecapole moments from measuring hyperfine splittings in the extremely long lived first excited state ($4f^{13}(^2\!F^{o})6s^2$, $J=7/2$) of $^{173}\mathrm{Yb}^+$. We present results of atomic structure calculations in support of the proposed measurements.