Does a cleric's alignment restriction extend to spells learned via multiclassing into an arcane spellcasting class?

No, that restriction only applies to cleric spells (or spells from other classes that have the same feature, e.g. druid). Casting an aligned spell is an act of that alignment, which could over time erode a cleric’s alignment and eventually put them on the outs with their deity. For example, a cleric/wizard of a good deity could cast an evil wizard spell, but that’s an evil act, and enough evil actions could turn them evil–and if they’re evil, they can’t be the cleric of a good deity, which means they can’t cast that deity’s spells any longer (see this question for how the cleric can start worshipping an evil deity instead).

Proving this is more difficult, however: the rules just say that a cleric can’t cast spells, no mention of only being restricted with respect to cleric spells. However, this is entirely consistent with how the rules were written—that is, it’s consistent with how the rules almost always assume a single-classed character, starting from 1st level. Multiclass characters, characters starting play at higher levels, those things are not the “norm” that the basic class descriptions cover.

We can see an explicit example of this with respect to specialist wizards—like clerics’ aligned spells, the restriction on specialists casting spells from certain schools is written just using “spells,” but Complete Arcane explicitly mentions that a sorcerer/wizard multiclass is one way you could get those spells back. See this answer for details.