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.
]Journey to the West Event Re-run
Hunting Quest Part 1
Fate/Accel Zero Order Advance Campaign
Fate/Accel Zero Order Event
Lostbelt No.1: Anastasia
5 Star Servants Tier List: Best Servants [4/13 updated!]
Best Servants [★4]: 4 Star Tier List
1-3 Star Servants Tier List
Best Craft Essence: Craft Essence Tier List
Ivan the Terrible (Ivan the Thunder Emperor)
FGO Reroll Tier List
Anastasia Nikolaevna Romanova
Irisviel (Dress of Heaven)
Ramesses II (Ozymandias)
Black Holy Grail (The Black Grail)
Talk on the Heated Sands
Nine-Headed Bull Demon King
Lunar Hot Springs