Finitely additive implies finitely superadditive (trivial), which implies arbitrary superadditive including countably superadditive (follows from monotone). However, arbitrary, or even just countably additive is not necessary true. An example is Banach finitely additive measure on power set of the plane, which exists thanks to Wallace–Bolyai–Gerwien theorem and Hahn-Banach theorem: it's finitely additive, but it's not countably additive because of Vitali sets.

Outer measure is countably subadditive by definition. Pre-measure is countably additive but not necessarily defined on a sigma-algebra, and when you try to extend it to an outer measure you end up with just countably subadditive.

Measure is countably additive by definition, hence countably super- and sub-additive.