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
Fox's Night Dream
Gentle Affection
Shaytan's Arm
Chiron
Hunting Quest Part 1
Machine Parts Gathering: Extreme
Secret Art: Pheasant Reversal
Ereshkigal
Arthur Pendragon(Prototype)
Gorgon
Black Holy Grail (The Black Grail)