Jay Rupp
Yeah this is definitely one of Batman's toughest match-ups. I still don't think I play it correctly so don't take my advice as gospel, but here is what I learned:
Never do naked jump ins or even cross-ups (unless you want to get cute and use his glide every once in a while). Bats are essential to this match-up, and if you do want to use them to jump in, try to make sure you have 3 bats out (I get anti-aired by his D2 if I release one bat and jump in). Use the bats for pressure, and try to mix things up as much as you can. His trait, range, and D2 are awful to deal with. His D2 gets under your 6 frame jab, so use B1 and especially, B2 a lot more. Your B23 string is pretty much the only useful string that has some range, so abuse it, but don't get predictable and throw some high/low mix ups.
A neat little trick that I saw REO do is use B1...followed by another B1 (he even followed it by a third one), instead of continuing the string. After B1, your opponent will most likely fuzzy guard the rest of the string in standing block in order to avoid the potential overhead ender, so doing another low is smart. Plus, even if he blocks it, B1 is neutral on block. If you're able to land two B1's in a row, your opponent will probably crouch block, and that will allow you to throw in your F3 overhead (which is useful in this match-up since it launches right away, but it's not quick enough to abuse since Aquaman has some pretty fast normals). You can MB it to be safe, but beware, Aquaman's trident scoop and trident rush will beat your armor since they have multiple hits.
Punish his D2 on block with B113, but keep an eye on his trait. If he activates it, don't finish the string, and instead go for a mix-up/more pressure.
Don't zone from full screen with batarangs alone, as you don't want to trade with his from the deep. If you want to zone (if you have a life lead or something), do it more from mid screen while using bats for help, and try to close in on him right after that. Use MB batarangs for frame advantage. I think this is one of those match-ups where Batman has to use meter. Try doing 112 into MB Batarang for a free jump ins (you don't get to jump in a lot in this match-up so take it while you can).
Aquaman's wake ups can all be beaten a jump in, so finish your combos with up grapple for a hard knockdown and try to jump in on him if you read a wake up attack. But again, keep in mind, some Aquaman players will just mash uppercut as they're waking up (they use it like it's a true wake-up attack, lol) so time it well.
Instant air glitter can help (make sure you don't mess up the input or you'll eat an anti-air) since Aquaman players are quick to mash D2 when they see a jump in. But make sure not to whiff it, as Aquaman can punish it with scoop as you're still in the air.
Most importantly, you need to find a way to get him to the corner, where your B23 string becomes annoying for him and you get to use the rest of your arsenal.
This is all I have for the moment. I play this match-up a lot and I still struggle immensely.
I believe
DarthArma once had a thread about this match-up, I wonder if he's been able to come up with some new tech/strategy against Aquaman.