It's hard to comment on a fight and revolves around the hypothetical skill level Jarvis would have after Batman trained him, because we don't really know how good he'll be. Didn't Alfred take down Owlman pretty easily? Or am I remember wrong?
I'm thinking Jarvis is too old amd Rick is too much of a weak noddle man, that there would need to be some pretty significant skill out put to make this a fight.