I think his kit is super fine as-is. His passive could be better but I guess it's okay. His Q is fine, his W could be a little more spicy and appealing, his E is definitely fine and is a defining part of him as a champion.
I think is ult is super weird.. He gets a LOT of damage from the passive but the active is really boring and makes him un-killable with little counter-play. He could have a lot of potential visually with his ult.
I think Riot should definitely rework Jax's ult, not necessarily completely change the fundamental aspect of it, but make it more exciting, visually appealing and interactive. (With more counter-play)
Active: Jax slams his lamp-post into the ground planting it and igniting a large circle around it for (z) seconds and becoming larger. He pulls out a large pole-arm to fight with.
While in the circle, Jax receives (x) armour and magic resist. He also deals (y) extra damage to champions.
Taking away the ultimate passive damage could leave room to possibly take a look at passive or W for potential changes. But I'm not a gameplay designer so who knows. (Just please, if you rework Jax, don't let CertainlyT near him.)
And more lore obviously!