I would skip the solo cup and straight from rockwool cube to final container.
And I would skip the rockwool and germinate auto flower seeds in the pot they'll be harvested in. Soak the perlite in the 1.5 gallon hempy pot with 1/4 strength nutrients. Push a small hole with a toothpick or something similar about 1/4" deep, drop in your seed, cover it up, and as a rule in 3 days it will sprout. That is a common germination time. Some can take a little longer but usually right around 3 days.