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In English, words like cleanliness or breakfast have preserved an older vowel than those in the free forms clean and break. In Japanese, compound noun accent tends to match between dialects, even though the individual components differ unpredictably (and moreover, the common compound position matches Ramsay’s reconstruction of Middle Japanese tones, so it’s likely to be a remnant).
Is there a general phenomenon where bound forms are relatively more resistant to sound change than free forms?
I don't think that bound forms tend to resist sound changes in general.
Bound forms might in some cases provide more information about the historical form of a word because they occur in a different phonological environment than free forms. For example, if a language has a sound change that eliminates word-initial or word-final consonants, than bound forms in compounds might show the historical presence of a consonant that was lost in the free form of the morpheme.
I don't know about Japanese, so I'll only address the English example. I don't think that it strongly supports the idea that bound forms are more resistant to sound changes.
English cleanliness and breakfast actually don't entirely preserve what is thought to be the historical value of the vowel: clean and break are supposed to have had long vowels in Middle English, something like /ɛː/. But cleanliness and breakfast* have the "short" vowel /ɛ/ in present-day English.
These are examples of a somewhat more general pattern where a morpheme may show a short vowel when it is used in compounds but a "long" vowel when it is used as a free form (and that in turn is part of a more general pattern in English where the frequency of "long" vowels relative to "short" vowels is higher in shorter words).
Other examples of shortening of an etymologically long vowel in compounds are vineyard, shepherd, and holiday (the last could be seen as an example of "trisyllabic" shortening, but the exact nature of that is a bit complicated).
Other examples of a "short" word having a long vowel while a related longer word has a short vowel: know, knowledge; south, southern; goose, gosling; please, pleasant; zeal, zealot.
For some reason, the values of short vowels seems to have changed less than the values of long vowels/diphthongs in the history of English.
*With break, the Middle English /ɛː/ was derived from "open syllable lengthening" of earlier short e, but I don't think that is directly related to the modern English pronunciation of breakfast. The OED's first citation for breakfast is from 1463 ("brekfast"), and I think that open syllable lengthening had already occurred by that time, although I'm not 100% sure.
You asked about resistance to "sound changes", but I wonder whether the different development of the position of the accent in compounds vs. non-compounds in Japanese could possibly have arisen from some kind of morphology-mediated effect.
I don't know whether it's relevant, but there seem to be some examples from Ancient Greek and modern Russian of compound adjectives being more morphologically conservative than simple adjectives. I learned about them from the discussion in this WordReference thread started by Scholiast: Another classical Greek oddity
It seems that in Classical Greek, prefixed adjectives are more likely to show an older declension pattern where the feminine form is identical to the masculine, as an o stem, rather than having a distinct stem ending in ā.
In post #2 in that thread, ahvalj compared this to a phenomenon in modern Russian where
derived simple adjectives always have a suffix, but compound adjectives often lack it, so that e. g. "four-footed" is четверо·ног·ий/četvʲero·nogʲ·ij, without a suffix, while a non-compound **ногий/nogʲij "footed" is impossible.
