Greetings and welcome to the jax qq thread where i flame the leaving sh*t out of riot and jax players. Jk let's proceed.
The passive part should be left untouched, that's what defines him as a 2v1 beast, an extremely strong splitpusher with consistently high dps and burst in a teamfight if left undisturbed. It is logical that it has no downtime. But let's take a look at the active defensive part of it, then let's compare it with other top laners' ultimate cooldowns.
Jax's active ultimate grants him
30 / 50 / 70 (+ 50% bonus AD)
Bonus Magic Resistance:
30 / 50 / 70 (+ 20% AP)
for the next 8 seconds, while not necessarily impactful in a manner, other ultimates are, like bringing aoe cc, aoe damage or survivability for the rest of the team, it does directly impact how many resources you have to waste in a teamfight in order to pin down/remove jax from it. That is on an 80 seconds cooldown at rank 1/2/3 without any cdr, an effective 72 seconds cooldown if you consider the fact that the ultimate goes on cooldown immediately and not after the effect ends. Assuming jax normally rushes sheen for an early 10% cdr, that's an effective 64 seconds cooldown which means that he can basically have it up everytime he comes back to lane.
Let's take a look at similar ultimates that grant defensive stats and their cooldowns.
Dr. Mundo 110 / 100 / 90
50 / 75 / 100% maximum health over 12 seconds. An effective 98 seconds downtime. 87 seconds with 10% cdr
Trundle COOLDOWN: 100 / 80 / 60
Total Magic Damage:»
20 / 27.5 / 35% (+ 2% per 100 AP) of the target's maximum health
40% mr and armor stolen over 4 seconds that remain stolen for another 4 seconds after the drain effect ends. 92 seconds effective cooldown, 82 seconds with 10% cdr
Renekton COOLDOWN: 120
250 / 500 / 750 Granted for 15 seconds, an effective 105 seconds cooldown, 93 seconds with 10% cdr.
Nasus COOLDOWN: 120
300 / 450 / 600
15 / 35 / 55
Total bonus resistances over 15 seconds 30 / 65 / 100. 105 seconds effective cooldown, 93 seconds with 10% cdr.
My point is that riot has dealt with low cooldown, highly impactful ultimates during laning phase in the past. Let's take riven for example.
Blade of the Exile
Cooldown increased to 110 / 80 / 50 seconds from 75 / 60 / 45.
Riven used to be able to dumpster you once she hit level 6, push the wave, recall, enter the lane and all-in you over and over and over again ( a never ending cycle).
Her ultimate received further cooldown nerfs because they deemed she still had it up way too frequently
Blade of the Exile Blade of the Exile
Cooldown increased to 130 / 95 / 60 seconds from 110 / 85 / 50.
My suggestion is that you issue a downscaled cooldown per rank on jax's ultimate, with the nerf mainly targeting his rank 1 ultimate, while receiving small compensation in the late game where he is supposed to shine. Something like 100/85/70. This should allow for meaningful trade windows in the laning phase against him.