You can't, neither your weapon nor the strider's weapon are equiped while mounted. Without a weapon, no SS or SpS. Their skills can only be used when you're not mounted.
However, fighting from a strider does give you a number of substantial advantages over fighting unmounted. That is, if you have a fighting strider, not one of those twilights that are easly to level and as common as Honda Civics.
ya I have the nuker strider and did notice that my mattack is higher mounted than dismounted even though I am lvl 69 and he is still only 55 (I lvled him myself)