Keebler,
I can't comment with any weight on how much running naval officers do, however I think you are very safe in your assessment that they run far less than Combat Arms/personnel posted to Army units. Even in Cmbt Arms units, they don't "just run" to stay fit. There are many threads on the forum about fitness, however I am not sure many are on fitness "norms" in the Navy...might be worth posting on Navy thread if you are really wanting to get info "from the horses mouth".
The EXPRES test is the current "CF" standard fitness test for non-army units. Note, it is not so much the uniform you wear as it is the unit/formation you are posted to. Units belonging to CLS (The Army) and all personnel posted to these units MUST do the BFT (Battle Fitness Test) which might be correctly called the AFS (Army Fitness Standard) Test? Either or, IF you were posted to an Army unit, then the AFS applies to you (for example, a Supply Tech in the Navy DEU posted to a Service Battalion would have to perform to the Army standard. Once said Supply Tech is posted to say, MARLANT, they would then have to achieve the EXPRES test standard.
Lots of info on both, just search.