As seen in the description of Critical hits (PHB p.196):
Critical Hits
When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together. Then add any relevant modifiers as normal. To speed up play, you can roll all the damage dice at once.
For example, if you score a critical hit with a dagger, roll 2d4 for the damage, rather than 1d4, and then add your relevant ability modifier. If the attack involves other damage dice, such as from the rogue’s Sneak Attack feature, you roll those dice twice as well.
So this means an attacks damage rolls are double, but secondary effects, such as a spiders poison, are not. This extra d8 is added to the initial damage of the attack, and should thus be treated as a critical roll: you would roll 2d8 and add the number to one of your damage rolls, instead of just 1d8.