As soon as they break the surface of the soil, they can go into full sun, and all day every day.
Have been doing exactly like this for 35+ years and it really only imitates what happens in nature.
Those clear plastic cups are a bad choice for starting plants, there is barely any soil, meaning root growth will be severely limited, which will stunt your plants if left longer than 7 days, also, roots need complete darkness to grow properly, something that is not achievable in clear plastic tiny cups.
I have found black plastic pots from a garden centre, about 4 inches across and 5 inches deep are close to ideal for 2-3 weeks growth.
Plus, if you look after them and wash them between grows, they will last many years, making them quite economical in the long run.
Hope this helps,............
Organoman.