You mention that it seems to have moss on it but given the distance the photo was taken from the tree I am having difficulty seeing it. You may send a close-up photo if you wish. What you see could possibly be moss or lichen. More information on moss & lichen can be found here:
Your barberry does look a little tall and scraggly. Barberry bushes do respond well to rejuvenation pruning so that it something you may want to consider. More information on rejuvenation pruning can be found here:
As for fertilizer, you can apply an all-purpose, slow-acting granular fertilizer.
For more information on the care of barberry bushes, please visit the following link: