For a vibration in the string, the speed depends on the mass and tension on the string (it's a transverse wave). It will be less than c.
For a pull, the string will stretch, and then the wave propagates like sound (longitudinal wave) and will move at the speed of sound in the media.
BTW, doesn't matter if you replace the string with a neutronium rod. Still takes time (more than light) to propagate.
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