Usage of 的: The first day of (or without of) a fast? Is correct to translate "The first day of a fast" (fast — in religious meaning) as: 第一天快? Or the better to use 的: 第一天的快?
When 'fast' refers to no food/drink/both/[sometimes sex] for a period of time by religion (Ramadan, for example), it's "斋戒."

When 'fast' refers to no food/drink/both for a period of time for political reasons (hunger strike), it's "绝食/绝食抗议."

When 'fast' refers to no food/drink/both for a period of time for medical purposes (blood test/surgery), it's "禁食 (verb)/空腹（adj.)."

"快" means "quick/fast/high speed" in adj.  "about to/nearly" in adv.

In you case, I would say "斋戒(的)第一天。"