There were quite a few Chinese typewriters, all trying to figure out the best solution. Thomas Mullaney writes about what he's called the "Typing Rebellion" in China. His point is that the west has long held that opinion of the Chinese typewriter. Several dozen variations of Chinese typewriters emerged in the last century, all trying to make a better solution, becoming so advanced that they blended with the emergence of the computer. The short blog post above is worth reading.

Japan made use of katakana, one of their syllabaries, for telegrams. It had only 47 characters, so could be conveyed through a variation of Morse code. You can see an example here.

The answer to the other half of your question is that they (the Japanese at least) just made extensive use of handwritten documents. As another poster has mentioned, typewriters capable of handling Chinese characters were fairly widespread but were labor-intensive enough to not be used for fairly trivial matters (although Japanese typewriters had the advantage of including less than half the number of characters).

I've done research in Japanese and American archives for government documents on the same event and while American documents are almost without exception typed, the majority of the Japanese ones are handwritten. Important documents such as interministerial requests and final reports are the exception.

Last week someone asked a similar question, which I responded to at some length. See my second response for typewriters specifically.

Telegraphs didn't come up there, however, China also had telegraph codes for the characters which could be used to transmit messages.

Here's an example of some. Note that some of the circled characters there are just components of more complex characters, so the person doing the transmitting could compose a character with an unfamiliar code using much more familiar components. This could take more time to transmit, but potentially less to compose (vs looking it up) and understand on the other end.