We employ the boundary integral method to compute the enhancement of spontaneous emission and the antenna efficiency for a quantum emitter coupled to a plasmonic core-shell nanocone. We consider the classical model of a dipole in the vicinity of a nanostructure and solve Maxwell’s equations to study the effect of shell thickness and core refractive index on the total decay rate and the antenna efficiency. We also investigate the influence of position and orientation of the emitter, which are not easy to control from a practical point of view, in order to achieve a better understanding between experimental results and theoretical calculations. It is shown that by adjusting all mentioned parameters, the spontaneous emission rate is considerably enhanced. Moreover, one can easily tune the localized surface plasmon resonance in the visible and near-infrared region, where absorption in gold is smaller, by changing the shell thickness or the core refractive index