Undead Fortitude. If damage reduces the zombie 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 drops to 1 hit point instead.

Infection.  A humanoid bitten by a zombie makes a DC 13 Con save.  On fail they lose 1d6 max hit points and lose 1d6 to their HP max per day.  A bitten humanoid who dies rises 1d4 hours later as a zombie, unless the humanoid is restored to life or its body destroyed.  Spells like Greater Restoration, Remove Disease, or Wish can remove the infection.  

Spellcasting. The Zombie Mage is a 1st-level spellcaster. Its spellcasting ability is Wisdom (spell save DC 12, +4 to hit with spell attacks). The Zombie Mage has following spells prepared:

Cantrips (at will): Chill Touch, Ray of Frost, thaumaturgy

1st level (3 slots): Shield, False Life, Thunderwave

2nd level (1 slot): Ray of Enfeeblement

Slam. Melee Weapon Attack: +3 to hit, reach 5 ft., one target. Hit: 4 (1d6 + 1) bludgeoning damage

Bite. Melee Weapon Attack: +3 to hit, reach 5 ft., one target. Hit:  4 (1d6 + 1) Piercing damage.  *See Special Trait: Infection*