Were Archimedes screws used in pre-modern times in the far east?

Evidence of the Archimedes screw pump being used in Japan first appears in 1618, though the idea of a screw had been introduced around the middle of the previous century. In China, it is mentioned in books from the 1620s.

Note: The Japanese medieval period is given as 1185 to the mid 16th century by the 'Cambridge History of Japan' but up to 1603 by Wikipedia. For China, the terms 'medieval' and 'middle ages' are not typically used: those who do have imposed various end dates.

JAPAN

The earliest mention of any kind of screw in Japan would appear to be 1544:

On another ship from China, arriving in 1544, was a Portuguese blacksmith who taught Yaita how to apply a screw at the bottom end of the barrel. With this screw in place, the risk of explosion was eliminated.

For the Archimedes screw pump, the earliest mention and use I've found is 1618 where it was used in the Sado gold and silver mines to deal with flooding:

The Archimedes screw pump (suijorin...) was first introduced in 1618, in an effort to tackle the perennial problem of flooding, and this heralded Sado's most productive period.

The Archimedes screw was also widely used on Sado from the mid seventeenth century...

Sado mining scrolls depiction of the Archimedes screw. Source: sado-goldmine.jp

It is also mentioned in the text of this British Library article:

Where no running water was available a pit was dug on the beach and filled with water. An Archimedes screw is shown drawing the water up out of the pit to form a flow that could be used for sluicing.

CHINA

(All quoted text below is from Archimedean Mechanical Knowledge in 17th Century China by Zhang Baichun and Tian Miao, in The Genius of Archimedes – 23 Centuries of Influence on Mathematics, Science and Engineering)

In China,

In 1612, Archimedean-screw was introduced by Sabbathinus de Ursis and Xu guangqi.

The above refers to the work Taixi Shuifa (Hydraulic Methods of the Great West, Hydraulic Methods). Sabatino de Ursis (1575–1620) was an Italian Jesuit who worked with the Chinese scholar Xu Guangqi to translate western texts into Chinese. This work "depicted some water-lifting devices such as Archimedean screw". In the book,

Xu Guangqi, the Chinese partner in translation of the book vividly named it [water-screw] longwei che (vehicle with the shape as dragon’s tail)

The book also has an illustration of a water-screw. Then,

In 1623, Italian missionary Giulio Aleni (1582–1649), assisted by Chinese scholar Yang Tingjun, compiled Zhifang Wai Ji (Areas Outside the Concern of the Chinese Imperial Geographer, ZFWJ),...In which Archimedes and his inventions were mentioned...

It was also mentioned in Diagrams and explanations of the wonderful machines of the Far West (Yuǎnxī qíqì túshuō lùzuì), printed in 1627:

The third chapter...illustrates, and shortly describes,...cranes, mills, water-pumps, the Archimedean screw and (for the literati!) a revolving bookcase.

Source: Peter M. Engelfriet, 'Euclid in China: The Genesis of the First Chinese Translation of Euclid's Elements, Books I-VI (Jihe Yuanben, Beijing, 1607) and Its Reception Up to 1723' (1998)

However, these tomes do not appear to have been widely available and the Archimedes water-screw mostly remained in books. There are a few examples, though, of manufacture and practical usage:

There were some accounts of manufacturing and use of the waterscrew in 18th and 19th centuries. Xu Chaojun...had a good grasp of astronomy and clockmaking. A book, which printed before 1911, told us that he constructed a water-screw that could be driven by a child to irrigate crop in 1809.

Qi Yanhuai (1774–1841), who first held office as a county magistrate in Jinkui and afterwards as a prefect in Suzhou, also made a water-screw and a pump

Source: Zhang Baichun and Tian Miao

Mostly, though, it was found to be impractical and expensive:

For craftsmen and farmers, the construction and using of water-screw also bring some inconvenience. Qian Yong told us a short story. A waterscrew was made in Suzhou in 18th century. It may irrigate cropland of thirty or forty mu (a traditional unit of area) every day. However, its very expensive in construction. And if it was damaged, it could not be repaired and be used again. A majority of farmers was so poor that they were not able to make it...

Source: Zhang Baichun and Tian Miao

To summarize,

...Archimedean mechanical knowledge exerted a limited influence on China in 17th century. It had partly been studied and practiced by Chinese by the mid-19th century.

Source: Zhang Baichun and Tian Miao

KOREA

References to Archimedes in Korea before the 20th century are few and far between.

The closest reference I've found is 1783 when the Joseon Dynasty King Jeongjo (ruled 1776 - 1800) was advised to use a

Yongmi-cha (Archimedean screw), a type of western waterwheel

However,

though many Joseon scholars were fascinated by the enlightened ideas of the West, none of the theoretical designs were actually used to construct real waterwheels.

It is possible that some among the Korean elite were familiar with Chinese works which mentioned the Archimedes water-screw. The Korean scholar Choe Han-gi, in his 1867 book Seonggi unhwa, mentions several "contributions to physics" credited to Archimedes but not the water-screw.