Originally Posted by
Zynetic
To answer your original question;
Doing field runs will increase drop rates by 25%, while doing quests will increase experience gain by 25%.
Of course, since this weapon drops from the second half of Shrine, it might be faster to do the quest that runs you through the second half of Shrine, as the time taken to get there in a field run might make the 25% extra drop rate meaningless. (although, I'd run full runs since you can get cool stuff in the first half of shrine)