How much real time do I have to wait to finish the fifth Epsilon Program mission, Bearing the Truth?

One in game day (24 hours) is 48 minutes in real life so you have to wait 8 hours in real life.

You can either wait it out as suggested in this other answer or you could quickly finish the mission by repeatedly sleeping on a bed, which will advance in-game time.

From the IGN GTA V Wiki Guide for the Kifflom! trophy/achievement:

After the robes have been delivered (takes 1 day), equip them from Michael's closet. If they don't show up, try quitting and re-entering the game. You need to wear them for 10 cumulative in-game days (6 real-life hours). When you save the game using Michael's bed, it progresses the in-game time 6 hours, so the fastest way to do this is to sleep 40 times... An indicator will appear in the bottom-left of the screen saying how many days you've worn them as you progress.

Alternatively, you can repeatedly kill yourself as Michael (this is faster but this will cost you money for the hospital bills). From the GTA Wiki's Bearing the Truth article:

...the fastest possible solution for finishing this mission quickly without risking a wardrobe change is to repeatedly kill yourself as Michael, with the robes on. With every death, 12 hours will pass in the hospital, meaning you will only need to take 20 visits to the hospital in order to complete the mission. However, each hospital visit will cost $5,000 so it's advised to use all money on Michael for property or stocks.