Helping/Abandoning Arie van Bruggen
When you turn over the Tai Yong keycard to Van Bruggen, he'll churn out a hacked version with your information on it after a nifty little cutscene. This cutscene also triggers a fairly massive raid from Belltower mercenaries that are determined to kill every last thing in Alice Gardens as part of the cover-up.
As noted before, try to keep a throwaway gun on you for when you bring the keycard to Van Bruggen. Once the Tai Yong Medical mercenaries storm Alice Gardens, you'll be given a short choice.
Van Bruggen will ask that you hand him a weapon so that he can at least have a chance to fight if they find him. This is all up to you. You'll be able to hand him just about anything that you have through the conversation wheel that comes up, or you can refuse. Note that refusing is explicitly a "you're on your own" message. Hopefully you have a nice cheap unmodded handgun for him. It doesn't matter what you give him, since he plans to hide in his pod regardless. Any gun will work. It's really just for show.
If you don't mind spoilers, note that there's a fairly practical side. You'll get a nice bonus of 2000 credits a little later in the game if you hand him a gun, so it's a fairly good deal.
Alice Gardens Courtyard
As for the way out, note that the courtyard is full of guards and heavily armed mercenaries. Note that several in the main courtyard have heavy rifles and armor, so while combat is still possible, you'll want to be careful about exposing yourself to fire from the ground.
If you want to just fight them, sneak over and quickly kill the two men on your level. They're in a side room by themselves, so it shouldn't be hard to gun them down or do two quick stealth takedowns. Then just use the steel railings as cover to take the rest out strategically. The heavies will stay on the ground floor, so raising an alarm will nicely split them up. You can then just pick them off on the stairs and take out the armored men last.
If you want a more strategic path, there's a easy way out. Most of the men are in the courtyard. You should be able to just slip past the two in the pod section next to you, or slip inside and chain up two takedowns after they split up.
Your goal is not to go through the front door. There's an easily accessible vent in section 1, so your goal is to get down a level and slip inside and to the back to reach that vent. You can then just climb through and you'll pop out past the courtyard close to the manager's office. Once you make it past the two men on your floor, it should be a simple exercise in staying crouched and moving forward while the lower guards aren't looking.
The Way Out
The main exit is sealed, so you need to cut through the laundry and shower areas in the basement.
The path is a fairly standard set of patrols. You'll find pairs of guards, so feel free to be aggressive and just use takedowns and non-lethal weapons to quickly cut through them. Combat is also fairly easy, since you should start any fight out with a shot to the head. Then it's just a matter of gunning down the survivor.
The first pair of guards will be in a locker room. Just use the lockers for cover and take them out. The second set is patrolling a narrow corridor, which is also covered by a turret. You can just ghost past them if you want. There's a vent in the side wall that will put you out behind the turret. If you wish, you can also sneak behind them and hack the panel to turn the turret against them. it's up to you.
After two pairs, you'll reach a slightly more dangerous area. This set of shower stalls is a bit of a killing field. There are two paths through the area. The main path is electrified. With the right skin augmentations, you can just walk through it. Otherwise, you'll need to go to the right…which has a bunch of flashing lights. It turns out that those are hidden mines, in this case, fragmentaton mines. These will kill you in one blast, even on normal. You need to grab a crate and throw it into the start of the room to set off a ton of explosives. If you want a bunch of fragmenation mines, you can also crouch walk (not run) through the showers and disarm them. That's not as fun though. The second set of mines can also be set off with one good throw.
Once that path is clear, just go through. You'll end up in the laundry room, so quickly take cover and move up to take out another two guards. You should have no trouble just putting bullets or tranquilizers into them thanks to the corridor. When they're down, just move forward and hit the exit to escape the ambush.
Note that from this point on there will be Belltower guards around the entrance to Alice Gardens. You should be careful about hacking or doing any illegal actions around them.
- All information and screenshots from Deus Ex: Human Revolution