We demonstrate theoretically that it is possible to manipulate electron or hole spins all optically in semiconducting carbon nanotubes. The scheme that we propose is based on the spin-orbit interaction that was recently measured experimentally; we show that this interaction, together with an external magnetic field, can be used to achieve optical electron-spin state preparation with a fidelity exceeding 99%. Our results also imply that it is possible to implement coherent spin rotation and measurement using laser fields linearly polarized along the nanotube axis, as well as to convert spin qubits into time-bin photonic qubits. We expect that our findings will open up new avenues for exploring spin physics in one-dimensional systems.