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.
5 Star Servants Tier List: Best Servants [5★]
Best Servants [★4]: 4 Star Tier List
1-3 Star Servants Tier List
Valentine event: Chocolate Lady's Commotion [US 2018]
Nero Claudius (Bride)
FGO Reroll Tier List
Jack the Ripper
Talk on the Heated Sands
Street Choco Maid
Kitchen ☆ Patissiere
Holy Night Supper