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If a class feature gives you something, then you just get it, period; it doesn't matter what character level you get the feature at.
In other words, yes, multiclassing into wizard (or any other spellbook-using class) gets you a free spellbook, complete with whatever spells the class feature says it contains. Likewise, multiclassing into gunslinger gets you a free battered blunderbuss/musket/pistol.