No one is giving you a solid answer.
oh well, Jax has one of the best normals in the game and one of the best Anti-Air options as well, there is a few you can work on against crossups, jump ins or interrupt stagger strings if you can read them.
option1: Jax D1 works well against crossup if you wish do use this option without using any meter its advisable.
Option 2: Reverse the inputs from Gotcha air grab, to catch them on the opose direction when they go for the crossup, i have no confirmation if that is a good option tho, but is still an option.
Option3: Ex Overhead Smash has armor, you can stop anything that fall in crossup or ppl that tries to jump you over, stagger you over with dial strings like 3D1... D1 smoke with this special move alone, the problem is that it costs one bar of meter, but is nearly guaranteed, specially if they are not expecting or are already air bone.
option 4: Jump Backwards with a kick and grab in the air, most of the time this works.
Don't get me wrong, but I was maining Major Briggs and I need to inform on the anti-air game you mentioned as I don't know where to start or how much he knows about Jax.
Option 1: d1 is used on opponents who press their jip (jump in punch) too soon, better ones won't get hit by d1 or start kicking. So use d1 with caution or not at all (don't be greedy with dmg)
Option 2: if you stay still the jump is like 40 frames or so and air gotcha grab is 15. A mediocre cross up will reverse your inputs at 20 frames (half jump) and you have to react at frames 21-23 max because the move is too high and will whiff. You get your invincibility to air attacks though, but there is chance to still be punished once your opponent hits the ground.
In a good cross up (those which cross up last moment) the move won't even be executed.
UNLESS...
you dash out, then air gotcha grab. And that's the most solid option, your move comes out faster and you are out of range for most quick moves.
(I didn't mention it at first cause I wanted to share info, I miss MK9)
Option 3: overhead smash is good, but it's 28 frames and can be baited with empty jump, jip teleport and even jip into armor upon landing sometimes. The feeling for using a bar for nothing...
There is another move for using a bar.
EX superman punch (yeah that's what I call it) 13 frames, practically same dmg, corner control and teleport evade.
Option 4: What is th... no, that's good it works