Two similar but different situations occur depending on whether you're in Survival or Creative.

Expectation:
For the jockey to become a regular skeleton upon death of the spider.

Occurrence:
Creative:
After killing the spider the jockey glitches through the floor. His hitbox is still with the model and the arrows come from relatively the same area as well. Once, the model was out of sync about 7 blocks from the hitbox and the arrows shooting.

Survival:
After killing the spider the jockey appears to glitch through the floor, but becomes invisible. While testing I saw two skeletons glitched through the ground about 25 blocks away from where the jockey was killed. They were invincible and didn't attack. The skeleton ended up falling into a creeper hole at which point he became visible and attack-able.
The invisible jockey's arrows were visible and did damage. When testing a spider was hit and the spider attacked the jockey and did damage (the noise was heard).