Yes, all of these cultures expect and use sorting pretty much just like alphabet-using cultures do.
Japanese has a set of some 46 phonetic characters called kana. They're arranged by phonetics in a table of fixed order, called the 50 sounds table (gojūonzu), a descendant of Sanskrit phonetic tables. Textual data, glossaries, lists etc. are usually sorted in gojūonzu order, and speakers will mentally use gojūonzu order to find words in indexes, etc. There's also another system called iroha order. In the past, people learned to write with a poem called the Iroha song, which uses every kana symbol exactly once. The order of characters in the poem then doubled as an order for sorting. This is today old-fashioned, but you can still stumble on things sorted by iroha order from time to time.
For counting, Chinese had a set of 12 ordinals called the Earthly Branches (dìzhī, 地支), and a separate set of 10 ordinals called the the Celestial Stems (tiāngān, 天干). These are thought to derive from the 12 months of the lunar year and the 10 days of an ancient week system; they're used to count things like "A, B, C…" or "I, II, III…" in lists. (The 12 Branches were later associated with 12 Zodiac-like animals, providing an easy mnemonic.) By combining one Earthly Branch and one Celestial Stem to make a pair, and then iterating both of them together as if in a system of gears, you'll get an ordered sequence of 60 pairs; this is used for counting years in the sexagenary cycle. The Japanese use these Chinese counters for sequences, too.
Sorting Chinese text itself, like we sort text in alphabetical order, is trickier. Some dictionaries and things today sort by pronunciation, arranged in alphabetical order according to its pīnyīn spelling in the Latin alphabet. (That is, first convert Chinese characters to the pīnyīn alphabet, then use that spelling to sort them.) Older systems generally order characters by number of strokes, and by the components they have in common. By far the most used system for that is the list of 214 Kangxi section headers, originally the indexers of a dictionary dating from 1615.
In computing, the sort order of text is called "collation", and there are standard algorithms to sort text in the order expected by each language/script.