Under dielectric continuum approximation, interface optical (IO) and surface optical (SO) phonon modes as well as the corresponding Fr\"{o}hlich electron-phonon interaction Hamiltonian in a free-standing cylindrical quantum-well wire (QWW) are derived and studied. Numerical calculations on GaAs/Al${}_x$Ga${}_{1-x}$As cylindrical QWW are performed. Results reveal that there are two branches of IO phonon modes and one branch of SO phonon mode, and the dispersion frequencies of IO or SO phonon modes sensitively depend on the Al mole fraction $x$ in Al${}_x$Ga${}_{1-x}$As material and the wave-vector in $z$ direction, $k_z$. With the increasing of $k_z$ and quantum number $m$, the frequency of each IO mode approaches one of the two frequency values of the single GaAs/Al${}_x$Ga${}_{1-x}$As heterostructure, and the electrostatic potential distribution of the phonon mode tends to be more and more localized at a certain interface or surface, meanwhile, the coupling between the electron-IO and -SO phonons becomes weaker.