Kanji (which derive from Chinese characters) are indexed based on the on-yomi reading which approximately matches the Chinese sound - albeit as understood several centuries ago. The index can be ordered on gojuon ('fifty sounds') or Iroha, a Japanese poem which contains all syllabic sounds of the Japanese language exactly once. This is more likely as Iroha was the standard ordering for centuries and continues to be used today in many contexts.

https://en.wikipedia.org/wiki/Iroha

Japanese is hard to read and write, but you can do it with practice. Complex kanji are not unlike complex words whose spelling you can learn through chunking. There are ~214 bushu, aka 'radicals', fairly simple characters from which all more complicated characters are made. In practice you only need to know about half of them unless you plan on being a hardcore language scholar, the others are rare. Once you get used to this, complex kanji are not arbitrary pictograms of 5-30 unique strokes, but small arrays of 2-4 bushu which are combined and written according to a small (8,8) set of rules. I think this is way more useful than the mnemonic approach used in many popular books.

In short, if you want to learn written Japanese (or Chinese/Chinese adjacent languages), learn the radicals early on. You can find Anki decks to help and mastering 214 of them is reasonably easy. Also, write. Nothing improves your recognition/retention like writing them regularly with a pen or a brush. Write them on your hand. Write them in the air.

They are ordered based on the readings: https://www.thisiscolossal.com/2021/04/typewriter-three-lang...

Apparently 1,172 characters on that machine. No mention I could find of composability of characters, which is possible in theory (many characters are combinations of simpler characters, but combining them alters the representation, shrinking the component characters/substituting slightly different marks).

Something else I didn't see mention of: can you easily switch the cylinder or the rows to get different character sets? I can imagine a business making due with ~1,000 characters because their subject matter is narrower than a newspaper would generally need.

I suspect you'd memorise at least the common ones with enough practice. After all, there are people who have memorised a significant fraction of what could be considered equivalent to Unicode codepoints: https://en.wikipedia.org/wiki/Chinese_telegraph_code

The possibly unique instance of the MingKwai typewriter, thought lost, was recently found, by the way:

https://typewriterrevolution.com/the-discovery-of-lin-yutang...

The tray-based design not only permits swapping out the entire characterset, but also sections.

My understanding is that Chinese typewriters typically have a large number of characters in common, but also provide for characters specific to specialised activities, e.g., government, commerce, education, science, technology, and the like, which are addressed in the auxiliary characters used.

Vague recollections from a New Books Network episode some time back. I'm pretty sure it's this episode with Tom Mullaney:

<https://newbooksnetwork.com/006-the-chinese-typewriter-a-his...>

Mullaney is also the basis for several other links provided by others in this thread, particularly <https://news.ycombinator.com/item?id=43212063>.