Scholars disagree about the exact origin of this relatively recent Bulgarian idiom. Some claim that the priest in question was an important man in Istanbul in the 19th Century, who even had connections in the Supreme Porte. Those wishing to express their dissatisfaction with the inefficiency of the Ottoman bureaucracy would use him as an intermediary for their interests; a simple call from the Armenian priest was said to be more helpful than dozens of official letters to the various levels of the administration.
However, the priest eventually died. Some linguists think that his passing may have given birth to the Bulgarian sayings "my priest died" or "my connections have gone to the cinema" (roughly equivalent to having "gone down the drain"), but that remains largely unproved since the expression exists in all Balkan languages, although in slightly modified form, and is not limited to the followers of any particular religion. Besides, cinema had not yet been invented at that time.
The saying fell out of vogue for a while, but caught on again in the mid-20th Century when an Armenian priest in Plovdiv managed to convince his parishioners that he had good connections which might be "useful." It turned out that Father Kirkor (not his real name) was just a charlatan whose main objective was to stock up on the fresh eggs and tomatoes brought to him by the gullible people of Plovdiv.
Hence the modern usage of the saying: to take your problem to someone who will accept your gifts, but won't do anything to help you.
By kind permission of HIGHFLIGHTS, Bulgaria's Airport Magazine