VS. Doomsday basic notes:
...Man I hate this guy. But that's neither here nor there. First things first:
Upshoulder (DB+3) on block: Power punish with your best b+3 combo if trait is not active. You may never see this depending on how smart the Doomsday is, but there you are.
Regular shoulder (DF+3) on block: d+1~DB+2 -> combo if trait is not active.
Telestomp (DB+2): Dash under b+3 is most consistent. I would really advise against trying to get fancy with Water Shield absorb, as if you don't dash cancel or release quickly after absorbing the unblockable, you'll miss your chance at a b+3 combo for optimal damage. If the corner makes things awkward for you, you can use f+3 to start your combo instead for the same amount of damage.
EDIT 5/4/13: Almost forgot, you can also punish blocked sweep with d+1~DB+2 -> combo if you're close enough and Doomsday isn't traited.
With that out of the way...
This guy is going to be in your face. A LOT. So do yourself a big favor and HANG ON to your meter for pushblock and the occasional MB b+3. This guy has a ton of ways to be safe or even advantage in your face, so don't hit buttons unless you know for sure what's not safe with him (which isn't much).
Don't spend meter on combos unless they're going to finish a lifebar (or put him in a position where he has to gamble on wakeup or die to chip). Seriously. Don't even bother with 1-bar BnBs. You don't need damage nearly so bad as you need to GET HIM THE FUCK AWAY FROM YOU.
Favor high block because his fastest low is his sweep, which is good, but not NEARLY as threatening or rewarding as his legions of safe overheads. Most of his unsafe strings, he can cancel into DF+2 to make both safe and bullshit (more on that below). You NEED pushblock or MB f+3/b+3 to get this guy off your back, blocking alone WILL NOT cut it.
AVOID JUMPING BACK. Properly spaced regular shoulder AS WELL as up shoulder can anti-air your shit very frequently. You're not gonna win this match from the air anyway. So plant your feet and let him get his rushdown out of his system. Dude is like SF4 Bison on PCP.
DO NOT pushblock regular shoulder BY ITSELF. You not only miss out on a guaranteed punish (if armor isn't up), but also if Doomsday MBs his shoulder, he can cancel the pushblock and stay in your face, and you can't pushblock again because you're still blocking. A raw deal for Aquaman. So if he randomly shoulders, just chill and get ready for a punish if it's regular. If he MBs it to stay in, too bad, keep blocking. That's the nature of the beast in this stupid matchup.
Biggest and STUPIDEST gimmick you need to know about with this guy is his Earth Shake (DF+2) stomp move. Hits overhead and then low so fast you have to fuzzy guard. 18f startup, so you can't jump or backdash out of it if he buffered it from a blocked d+1. Dumb. But that's not the bullshit part.
The in-game frame data says Doomsday DF+2 is -9 on block. This is a complete and utter LIE. This move is COMPLETELY SAFE on block to ANYTHING Aquaman does. Superman's super cannot reversal punish this move. It is safe on block to EVERYTHING IN THE GAME. Your best bet is to (1) practice blocking it just in case, and then immediately (2) start learning from your friendly neighborhood Doomsday players when they are likely to use this move, and pushblock before they even get to use it (say from a random blocked d+1).
EDIT 5/4/13: According to I$AAC, this move is actually +1 on block. Pretty damn dumb if true. Regardless, it pushes Doomsday too far back for Aquaman to force a blocked d+1~DF+1 MB for counter chip. None of Doomsday's safe moves will reach you from that distance, either. Doomsday has to either get reckless or let you breathe if you block this move. IF you block it. Try your very best never to be in a situation to block this move.
Also, if you're bad at blocking this move, favor blocking the 1st hit (overhead) over the 2nd (low), because the 1st hit does a LOT more damage than the 2nd one.
EDIT: Forgot to mention, one silver lining on blocking DF+2 is that you're spaced pretty much perfectly that you can stop Doomsday on the ground with b+1 or in the air with d+2 depending on what he does next. So this really is a good time to breathe a bit and OBSERVE Doomsday. See whether he favors backing off or pushing his luck after that mixup. And if you get a good read, for the love of God, finish your combo and save your meter.
You can reversal super between the d+1 and the DF+2 if he does that string, but unless it's going to END THE MATCH, it is a FOOLISH waste of meter to do so. You threw away four precious pushblocks/ MB b+3s for 33% damage. You DESERVE whatever frame trap hell Doomsday puts you through when he inevitably gets back in.