At the bottom, go into stealth mode immediately. Just inside the hangar are two mechs having a conversation. Hug the wall to the left and sneak by them.
Bounty! On the raised platform directly behind the mechs is Bounty Card 16. Getting it is tough, as you'll almost undoubtedly alert the mechs. One way to handle it: Make a bee-line for the card, grab it, and let the mechs murder you. You'll still have the card when you wake up back in the elevator for another go.
As you move around the outside of the room, you'll almost immediately come to a railing. Hop it if you can do so without being seen and drop down. At the bottom is a vent you can climb through to slip past the mechs.
Bounty! You'll find Bounty Card 40 along the way.
At the other end is a ladder. Climb out, then go around the back of the crate nearby. From the back, step around to the raised platform just on the other side, closer to the mechs. Use Y to climb up.
Now turn around and jump to the top of the crates. Hop over the railing and head left to the door. Get inside, kill the merc that's waiting for you, and start pressing buttons.
Hit the console to raise the elevator and the other to power up the ship. Then leave the control room. You'll need to jump back onto the crates and onto the raised elevator while avoiding fire from the mechs. When you get on the raised elevator, press Y to redirect the turret.
Now jump back to the crates and back over the railing. Return to the control room and press the button that says "Test Weapons Systems." That'll blow a hole in the glass of the hangar, killing the mechs.
After atmosphere is restored in the room, leave the hangar again and turn left to find a locked door. Dacher is on the other side and will open it shortly, ending your mission. You'll find a nanomed machine just beyond if you need it. You'll find Revas just beyond.
Boss Fight: Revas
Quickly equip your favorite blade. Revas is brutally strong and fast, so keep away from her until you're ready for a real fight.
Firstly, watch out for Revas' stab attack. It's pretty much unblockable and does a world of hurt. It's her go-to move as well. Try to step back from it whenever possible.
Instead, look for her to attempt to slash you with wide attack, and block that. That'll knock her off her game enough for you to slice her up.
Repeat this process. When you've done enough damage or you're quick enough, you'll catch Revas and give her back the hairpin you took from her at the outset of your stay on the Dark Athena.
Achievement Alert! Defeating Revas nets you the Escape the Athena Achievement, once a cutscene has completed.