Confucian Messianism, Mencius, etc Just picked up Popular Religious Movements and Heterodox Sects in Chinese History by Hubert Seiwert (In collaboration with Ma Xisha) and was kind of caught aback by the idea of Confucian messianism. Here's what the book says:

Google Books Page 20 of the book:


  One of these elements, which gave Han Confucianism a particular
  religious flavour, was the expectation of a sage-emperor who would
  realize the coming of a new era in which the world will enjoy peace
  and prosperity. These ideas were not confined to the Confucians of
  the New Text School but were widespread among different schools of
  the Former Han. Especially the fangshi 方士 (“master of recipes,” “magicians”),
  who were closely related to the Huang-Lao teaching, promoted
  the idea of a golden age that would be inaugurated by an emperor
  who responds to the cosmic order by enacting the proper rituals and
  cultivating his personality.10 Confucian scholars of the Former Han
  shared these ideas. They seem to have been inspired by a prophecy
  already alluded to in the Book of Mencius, namely the appearance of a
  sage every five hundred years who would transmit the true teaching
  and thereby restore order to the world.11 The Han emperor Wudi
  apparently referred to this tradition when he deplored the fact that for
  five hundred years the erudites had not been able to bring back the
  principles of the ancient kings.12 Wudi regarded himself as the one
  who should reinstall the cosmic order and become the sage-emperor
  preordained by Heaven. This is the background of the various ritual
  measures that he enacted during his reign.13





  The expectation of a sage-emperor who would bring about the
  ideal state of the world that supposedly had existed in high antiquity
  occupied the thought of the New Text scholars. It was this same
  expectation on which Wang Mang relied when he styled himself as
  the fulfilment of this expectation. Thus, the historical and political
  theories of Former Han Confucianism contained an element that could
  properly be called Confucian messianism. This Confucian messianism, of
  course, was intimately related to the belief in prophecies and omens popular with the New Text School. Following the failure of Wang Mang’s rule it lost much of its appeal and was eventually suspended.
  After the Han, messianic expectations no longer played a prominent
  role within the Confucian tradition.14





  10 Cf. Anna K. Seidel, La divinisation de Lao Tseu dans le Taoïsme des Han (Publications
  de l’École Française d’Extrême Orient; 71), Paris: École Française d’Extrême-Orient,
  1969, p. 25.
  
  11 Mengzi yizhu 孟子譯注, 盡心章句, xia, Beijing: Zhonghua
  shuju, 1960, vol. 2, p. 344.
  
  12 Hanshu 漢書, by Ban Gu 班固, Beijing: Zhonghua shuju, 1962 (1975), j. 56, p.
  2496.
  
  13 Cf. E.B. Ord, State sacrifices in the Former Han dynasty according to the official histories,
  Ph.D. thesis, University of California, 1987, pp. 112 ff.




Mengzi talks of 尧 and 舜 a lot throughout 盡心章句 - I would imagine it has something to do with this...?




What exactly is the prophecy being spoken of here, i.e.: the appearance of a sage every five hundred years who would transmit the true teaching
and thereby restore order to the world in the original Chinese? 
Is there a name for this, so-called, prophecy? 
What is the Chinese for the sage-emperor? (圣王 (?) )
What is entailed in the idea of a golden age that would be inaugurated by an emperor who responds to the cosmic order by enacting the proper rituals and
cultivating his personality?
What did Wudi (武帝) do during his reign that is considered to fit into this prophecy? E.g.: Wudi regarded himself as the one who should reinstall the cosmic order and become the sage-emperor preordained by Heaven. This is the background of the various ritual measures that he enacted during his reign.13 (this question might be off topic for this site, sorry.)
From 《孟子·公孙丑下》:


  孟子去齐。充虞路问曰：「夫子若有不豫色然。前日虞闻诸夫子曰： 『君子不怨天，不尤人。』」 曰：「彼一时，此一时也。五百年必有王者兴，其间必有名世者。由周而来，七百有馀岁矣。以其数则过矣；以其时考之，则可矣。夫天未欲平治天下也；如欲平治天下，当今之世，舍我其谁也？ 吾何为不豫哉！」


Mengzi was a very boastful person. I think the prophecy (actually more of a theory) 五百年必有王者兴 was something he made up to elevate himself to the savior (the One) of his time. He further talked about this prophesy of the 500 year cycle in 《孟子·尽心下》:


  由尧舜至于汤，五百有余岁，若禹、皋陶，则见而知之。若汤，则闻而知之。由汤至于文王，五百有余岁，若伊尹、 莱朱，则见而知之；若文王，则闻而知之。由文王至于孔子，五百有余岁，若太公望、散宜生，则见而知之；若孔子，则闻而知之。由孔子而来至于今，百有余岁，去圣人之世若此其未远也。近圣人之居若此其甚也，然而无有乎尔，则亦无有乎尔！


Mengzi listed the following nine sage emperors (圣王): Yao, Shun, Yu, Tang, Wen Wang, Wu Wang, Zhou Gong, Kongzi (素王) and Mengzi himself (舍我其谁).


  What is entailed in the idea of a golden age that would be inaugurated by an emperor who responds to the cosmic order by enacting the proper rituals and cultivating his personality?


I don't know the exact source(s) it is referring to, but again in 《孟子·滕文公下》:


  天下之生久矣，一治一乱。


The idea of 一治一乱 is simple: a period of peace and prosperity follows (and predates) a period of war and chaos. Whenever a new dynasty is established, the new ruler would employ theories such as 五德终始说 to claim his righteousness to reign.

Bear in mind that China had just experienced a period of war in Jingdi's (the father of Wudi) time and was constantly under threat from neighboring forces, so whatever Wudi did, the prophesy was used to solidify his power. The use of the prophesy, however, was not limited to the rulers. Sima Qian employed the 500 year theory to express his ambition in his autobiography 《太史公自序》:


  先人有言：「自周公卒 五百岁 而有孔子。孔子卒后至于今 五百岁，有能绍明世，正易传，继春秋，本诗书礼乐之际？」意在斯乎！意在斯乎！小子何敢让焉。


Notice his tone of voice was basically the same as that of Mengzi: in the present age, 500 years after Confucius (or Wen Wang before him), who else other than the Grand Historian is capable of producing something big and everlasting?

We can see the profound influence the 500 year theory had on later Chinese intellectuals in poems such as


  梅花重压帽檐偏，曳杖行歌意欲仙。後 五百年 君记取，断无人似放翁颠。


by the Song poet Lu You, or


  满眼天机转化钧，天工人巧日争新。预支 五百年 新意，到了千年又觉陈。


by the Qing poet Zhao Yi.