We demonstrate time-resolved nonlinear extreme-ultraviolet absorption spectroscopy on multiply charged ions, here applied to the doubly charged neon ion, driven by a phase-locked sequence of two intense free-electron laser pulses. Absorption signatures of resonance lines due to 2p-3d bound-bound transitions between the spin-orbit multiplets ^{3}P_{0,1,2} and ^{3}D_{1,2,3} of the transiently produced doubly charged Ne^{2+} ion are revealed, with time-dependent spectral changes over a time-delay range of (2.4±0.3) fs. Furthermore, we observe 10-meV-scale spectral shifts of these resonances owing to the ac Stark effect. We use a time-dependent quantum model to explain the observations by an enhanced coupling of the ionic quantum states with the partially coherent free-electron laser radiation when the phase-locked pump and probe pulses precisely overlap in time.