i'm trying to start a campaign where a group are, for a variety of reasons, in the Renkaru arcology prior to Deus's takeover. Then the doors close and they are trapped inside. However, what kind of missions/adventures/quests/whatever would be apriopate to this setting? The typical capture a drone for cash or extrat this prisoner does not seem to fit so does anyone have any suggestions?

yummy evil....

Are you planning a single run to get the runners back out, or are you imagining the runners having multiple runs to accomplish the overall goal of getting out?

How about simply surviving and trying to get out? Preferably alive of course.

Well, since getting out in one piece is already enough of a challenge I assume that you need a run to get them to go inside.
I always find it nice if they are invited to meet a Johnson for some VERY profitable run in one of the restaurants in the mall which occupies most of the basement level of the arc.
They all go there, sit down around the table, the Johnson begins to let them in on details of the run and all of a sudden the lights go out and overall chaos ensues...
Of course they could also meet him somewhere further up in the arc, but if a Johnson invites his runners to his own office or suite he must be either really dumb or really powerfull...

Suh-weet! A "Behind Enemy Lines" campaign Shadowrun style.

Some ideas:

1) Establish a safe-haven within the arcology for the group. Locate a Deus blindspot. Locate and set-up tech to ensure the haven remains a blind-spot.

2) Supply Lines: PC's will need food, water, ammo and medical supplies -
each lending itself to a raid style run with the team locating the resource and
planning out how to take the resource.

3) Allies: arranging meets with other pockets of resistance, arranging joint
missions to succeed at some important objective - rescuing an as of yet
co-opted otaku who would be a valuable ally also comes to mind.

4) The escape: GM's choice here - will the group need to shut-down Deus to
get out of the arcology or can they eventually find a way out while Deus
remains in control?

Chicago Containment Zone in the heart of Seattle, substituting an unstable AI and its sensors/minions for unstable insect shamans and spirits. Doesn't take all that much to descend into complete anarchy and raw survival drive, does it? And if you, as the GM, don't have a wider plotline in mind right from the beginning, such campaigns are more than likely to ground firmly and remain on that level.

Since the arcology before the lock down was Renraku's north American HQ and swarming with security. any run in there would be a recon, looking to get info on the location, people or gear. The runners would have light weapons at best. another idea is that they are chasing someone affiliated with another corp who ducked into the arcology. Still only light weapons, but as all hell is breaking loose, the runners are still chasing after the man from Cross AT to finish their run, and then get out. "What do you mean the brief case was eaten by a tiger? Oh the one that was in the zoo and is now in that bar."

That's the fun about it!

When they are trapped in there while meeting a Johnson or doing recon for a run or whatever, they'll be packing a minimum load since there's no way they can smuggle in anything big. And then, when all hell breaks loose, they won't have all their neat toys. This is one of the main problems of the people inside the arc. Supplies! The little stuff that you find in stores or loot from bodies lying around (guards, etc) won't last forever. Maybe you find an assault rifle and a clip if you're lucky, ut that clip will last for no more than three or four encounters!
Same goes for food and water.

Where do you get your wounds treated after some messy fights? Where do you sleep? What are going to eat and drink? What are you going to do when you run out of ammo or out of charge for all those neat toys like communications and stuff? All the details are what is so thrilling about this scenario. Think the toilets still work?