Pressure-induced superconductivity in a spin-ladder cuprate Sr2Ca12Cu24O41 has not been studied on a microscopic level thus far although the superconductivity was already discovered in 1996. We have improved the high-pressure technique using a large high-quality crystal, and succeeded in studying the superconductivity using 63Cu nuclear magnetic resonance. We found that the anomalous metallic state reflecting the spin-ladder structure is realized and the superconductivity possesses an s-wave-like character in the meaning that a finite gap exists in the quasiparticle excitation: At a pressure of 3.5 GPa, we observed two excitation modes in the normal state from the relaxation rate T-11. One gives rise to an activation-type component in T-11, and the other T-linear component linking directly with the superconductivity. This gapless mode likely arises from free motion of holon-spinon bound states appearing by hole doping, and the pairing of them likely causes the superconductivity.