The Kapitza resistance (R-k) at the water-carbon nanotube (CNT) interface, with water on the inside of the nanotube, was investigated using molecular dynamics simulations. We propose a new equilibrium molecular dynamics (EMD) method, also valid in the weak flow regime, to determine the Kapitza resistance in a cylindrical nanoconfinement system where nonequilibrium molecular dynamics (NEMD) methods are not suitable. The proposed method is independent of the correlation time compared to Green-Kubo-based methods, which only work in short correlation time intervals. R-k between the CNT and the confined water strongly depends on the diameter of the nanotube and is found to decrease with an increase in the CNT diameter, the opposite to what is reported in the literature when water is on the outside of the nanotube. R-k is furthermore found to converge to the planar graphene surface value as the number of water molecules per unit surface area approaches the value in the graphene surface and a higher overlap of the vibrational spectrum. A slight increase in R-k with the addition of the number of CNT walls was observed, whereas the chirality and flow do not have any impact.