Hi,
Basil can be grown under florescent lights so long as you can keep the lights about 4-6 inches above the top of the plant and leave them on for 14-16 hours a day. As far as you new plant goes after potting and watering well it should start to recover. The light is important from an intensity and duration stand point that is why it is placed so close and left on so long. Too far away and not left on long enough results in plants that get tall and spindly.