Pool Cue needs to be equipped. Staff of Fats does not.
Also, each time you get a certain semi-rare increases pool skill, as does level of drunkeness. Though we'd have to guess exactly what the value of the effect might be although we know that optimal drunkeness is 11. We also do not know the limit as to how many times you can benefit from the semi-rare.
With all those unknowns I doubt the usefulness of supporting "Pool Skill" unless next month's IotM gives us a worthwhile and quantifiable reason. (Which is entirely possible.) At least I finally know for a fact that none of the shark jumper, lucky bottlecap nor lucky rabbit's foot will help at pool, despite rumors to the positive.