When $n < 4$, $A$ is isogenous to the Jacobian of a stable curve because the Torelli locus is dense. This should imply that $A$ is isogenous to the Jacobian of a smooth curve because $A$ is simple (at least if the CM-type is primitive).

If $n \geq 4$, then (conditional) results of Chai-Oort indicate that $A$ is not necessarily isogenous to the Jacobian of a curve. Tsimerman then proved this unconditionally. I would guess that the set of CM points whose isogeny orbit is disjoint from the Torelli locus is dense in $\mathcal{A_g}$, but I don't know.

When $n < 4$, $A$ is isogenous to the Jacobian of a stable curve because the Torelli locus is dense. This should imply that $A$ is isogenous to the Jacobian of a smooth curve because $A$ is simple (at least if the CM-type is primitive).

If $n \geq 4$, then conditional results of Chai-Oort (see their recent Annals paper) imply that $A$ is not necessarily isogenous to the Jacobian of a curve. Tsimerman then proved this unconditionally. I would guess that the set of CM points whose isogeny orbit is disjoint from the Torelli locus is dense in $\mathcal{A_g}$, but I don't know.