Spider Climb. The Zombie Spider can climb difficult surfaces, including upside down on ceilings, without needing to make an ability check.

Web Sense. While in contact with a web, the Zombie Spider knows the exact location of any other creature in contact with the same web.

Web Walker. The Zombie Spider ignores movement restrictions caused by webbing.

Sense Weakness. The Zombie Spider can sense a creature close to death and will focus it's attacks on them if possible without endangering itself.

Undead Fortitude. If damage reduces the Zombie Spider to 0 hit points, it must make a Constitution saving throw with a DC of 5 + the damage taken, unless the damage is radiant or from a critical hit. On a success, the Zombie Spider drops to 1 hit point instead.

Bite. Melee Weapon Attack: +5 to hit, reach 5 ft., one creature. Hit: 5 (2d4 + 2) piercing damage, and the target must make a DC 15 Constitution saving throw, taking 7 (2d6) necrotic damage on a failed save, or half as much damage on a successful one. 

If the necrotic damage reduces the target to 0 hit points, they immediately take one failed death save as a Zombie Spider forms itself inside them.  If the target fails three Death Saves, a new Zombie Spider emerges from the corpse and attacks the nearest live target.