You can unlock the Core Slot of a Level 100 Ultimate Weapon by using a Core Binder.

The Core Binder is an item that can be researched and crafted by collecting Void Abyss Metal Fragments.

Void Abyss Metal Fragments can be acquired from 400% Infiltration Operations and Intercept Battles (Void Abyss).

When you equip a Core, one of the predetermined options is selected and assigned for a chance. 

If you re-equip a Core in a slot that already has a Core, you can decide whether or not to keep the existing option.

You can see what options are available for each weapon on the Unlock Core or Equip Core screens.

When you select a Core Slot, you can filter and view only the options that apply to that slot type.

After selecting a Core Slot, you can also select a Core to equip, to get an idea of what values the Core can be granted based on its Stage.

A list of Cores corresponding to the selected slot type will appear, allowing you to select and equip a Core.

Re-equipping a slot already equipped with a Core will consume a Core Particle Disruptor.

The Core Particle Disruptor is an item that can be researched and crafted by collecting Void Abyss Metal Fragments.