Green Emperor Way is the graveyard area that surrounds the Imperial Palace. It's quite handy as a shortcut between districts and is the site of a couple of quests, including this one. You're looking for the spot where the sun touches the top of White Gold Tower at noon. As it just so happens, that's right next to the only round tomb in the whole place. So just wait around Prince Camorril's tomb until noon (weather doesn't matter) and a glowing map will appear on the tomb. Activate it to update your journal and provide you with a map marker on Lake Arrius (in eastern Cyrodiil, kind of northwest of Cheydinhal), the appropriately named "Lake Arrius Caverns".
It's not entirely necessary, but you can go back to Cloud Ruler to report in. Jauffre just tells you to get to the shrine. Martin will have some information on Mysterium Xarxes, though it's not something you couldn't pick up later. Baurus is back in armor and looking hale and hearty (assuming that you kept him alive in the last quest). He'll teach you some tricks that will boost your Blade, Block and Heavy Armor skills by a point each.
So off to the caverns you go. Unless you're been doing a bit of exploring on your own, there probably isn't a fast-travel marker anywhere near, so be prepared to hoof it for a while.
You must decide how you're going to handle getting into the shrine. You can either go in guns blazing (so to speak) or finesse your way in. Finesse will get you in much easier, but getting out will be harder because you won't have any of your gear. "Guns blazing" makes it much harder to get in (out is about the same, either way), but at least you'll have your stuff with you. Considering that these are the guys who pulled you out of your peaceful prison cell and dumped you into all of this mess, killing everyone in the complex might be considered a "good thing".
You have to make that decision when you approach the guard in the first chamber of the caverns. Starting a fight with him will turn everyone in the complex hostile. Giving him the proper password will get you into the complex. In the next room, you'll meet Harrow, the Shrine Warden. Assuming that you didn't kill the door guard, he'll ask for all of your equipment. If you accept, it will all be removed and you'll be clothed in Mythic Dawn robes. If you decline, he'll go hostile on you. If you killed the door guard, he'll already be hostile. Since Harrow is going to hang onto your gold until you eventually kill him, it's your call on how to handle it.
If you're finessing, just follow Harrow through the Shrine Antechamber to the shrine itself. Otherwise, hang a right when you get into the big room of the antechamber (the portcullis up ahead on your left can only be opened from the other side).
Once you're in the Shrine, by whatever method, you get to listen to Mankar Camoran's little speech. He is "essential" at this point, so you can't kill him. Believe me, I've tried and the best I've been able to do is knock him out for a few seconds. About the only thing I haven't tried is picking his pocket (since he's not of the Septim bloodline, I don't think he can equip the Amulet). I doubt that this would work and even if it did, it would undoubtedly break the rest of the Main Quest. So just accept the fact that you're not going to be able to retrieve the Amulet and get on with the rest of the quest.
Once Camoran has finished his little speech, he opens a portal to his pocket Paradise and is gone, taking the Amulet with him. If you're finessing, you'll be told to talk to Ruma Camoran. She'll instruct you to kill the prisoner, Jeelius, that's tied up below the statue of Dagon. If you comply with her instructions, everyone else in the chamber will just kind of wander off, leaving you alone with a dead body, Harrow, Ruma Camoran and a guard. If you refuse to comply, they'll know you're an imposter and will go hostile on you.
Regardless of which option you choose, everyone goes hostile on you when you pick up the Mysterium Xarxes from the altar (which is your true quest object). Again, because I'm a "snipe from the shadows" kind of player, I typically don't have to make the choice and can just slaughter everyone in the room (except Jeelius, of course). If you don't free him, though, he's going to die when Dagon's statue crumbles and falls on him after you pick up the Mysterium Xarxes (you might get smooshed, too, if you're too close).
Now it's time to fight your way out of the complex. If you finessed your way in, the best you're going to have is a Silver Dagger and whatever quest objects might be in your inventory. Harrow has your stuff, by the way, and will just follow you around until you do something that turns him hostile on you. The simplest approach that I've found to getting my stuff back was to lure Harrow back into a dark corner and backstab him. Remember my earlier comment about running into Umbra as a 1st level character? I went back later and got it (hehehe). Since it's a quest object, it wasn't removed from my inventory. I didn't have a Black Soul Gem, so the Soul Trap effect was kind of useless, but it's still a good sword and, at 6x damage, put him down in one swipe - no fuss, no muss, no bother. Same thing for Ruma and the guard. Be sure to collect Ruma's Staff of Fire. If you don't have one of your own, it's decent enough.
If your Sneak skill is good enough, you can sneak your way out of the complex without being detected. Nice if your combat skills are a little lacking. If your stealth isn't so hot, I've only got one suggestion: if it moves, kill it (except Jeelius, of course). Some of the Mythic Dawn have spells, usually Shock-based, so a shield spell of some sort will help a lot.
OK, lets suppose that you cleared the area around the altar and didn't kill Jeelius. He's going to try to make a break for it (wouldn't you?), but the guy is a freakin' monster-magnet. He doesn't know the first thing about "let's not be noticed" and flat out runs the whole way. Because of his speed, I don't think there is any way that you can run interference for him. But he knows the way out and when he encounters one of the Mythic Dawn, he'll retreat back to you, assuming you're following (which you should be). Just keep following, kill whatever gets in your way and keep throwing "Convalescence" spells on Jeelius. When you return to the Lake Arrius Caverns section, look for a T-handle on the wall near the chest and sleeping pallet. Turn it to open the secret door leading back to the caverns (Jeelius somehow manages to bypass this - maybe he walks through walls?). If you followed closely enough, Jeelius will be waiting for you outside of the Caverns and will tell you that you can find him at the Temple of the One in the Imperial City. When you have a chance, do so. He'll boost your Restoration, Mysticism and Alchemy skills by a point each.
Now return to Cloud Ruler Temple and break the news that the Amulet got away. Jauffre tells you to take the Mysterium Xarxes to Martin. Martin will be able to provide some more information on the Oblivion Gates and how to close them permanently. The Dragonfires have to be lit and that requires an Emperor with the Amulet of Kings. At the moment, you only have one (Martin) and can't get to the other. After you finish this conversation, you do not get another journal update (delivering the Mysterium Xarxes closed this quest). But Martin will make a parting comment telling you to talk to Jauffre about "those spies".
If you have not already done so, talk to Baurus (if he's still alive) for your skill boosts. Then head for the Imperial City to collect your skill boosts from Jeelius (if he's still alive). 10 "freebie" skill boosts (3 from Baurus, 3 from Jeelius and 4 from the Commentaries) - not a bad haul, I guess.
On a side note, and something that will not show up in your Journal, completing this quest caused an escalation in the Oblivion Crisis. There are now Oblivion Gates outside of all of the major cities, except possibly the Imperial City. You can go close them now (which will make a later quest much easier) or wait. I prefer to take care of them now, but that's just me.