America: Charlotte (Free Quest) is the best AP-saving & time-saving location to farm Void's Dust. You'll get around 6x Void's Dust by spending 200 AP.
Fuyuki (Singularity F): Unknown Coordinates (X-F) is the first location you unlock to farm Void's Dust.
Shadow Servants drops Void's Dust. So you can get Void's Dust from Daily Quests as well.
Here is a list of servants that need Void's Dust to perform Ascension and Skill Up.
Rashoumon event (Ibaraki Douji event)
Murder at the Kogetsukan
Inheritance of Glory
5 Star Servants Tier List: Best Servants [5/9 updated!]
Best Servants [★4]: 4 Star Tier List
FGO Reroll Tier List [5/11 updated!]
Mysterious Heroine X (Alter)
Victor from the Moon
Octuplet Crystals (Octuplet Twin Crystals)
Marie Antoinette (Rider)
Cu Chulainn (Lancer)
Gaius Julius Caesar
Piece of Year 2030 (A Fragment of 2030)
Dumplings Over Flowers