If we consider metric spaces to be categories enriched over $\mathbb R_{\geq 0}$, the object corresponding to presheaves should be lipschitzcontinuous functions $\operatorname{Lip^ 1}(M, \mathbb R_{\geq 0})$. Now there should be an obvious metric on this set; making the Yoneda map $$x\mapsto \operatorname d(,x)$$ an isometric embedding. What is this metric?

It is the usual sup metric. See section 2 of Lawvere's original article. 

