How to make best use of the urban atmosphere?
PC-NPC Connections. The PCs don't live on an island. They have background - draw on those and give them a chance to use their background features. Create ally NPCs that are
actually helpful and interesting (not just an excuse to be rescued, give quests, or betray the PCs). Include a bit of tension/friction with the dilemmas you set up; there's a whole gamut of conflict between "friendly ally" and "open hostility and combat."
Dynamic NPC Plans. There are a lot of power-brokers in a city, and they aren't just going to be sitting still while the PCs mess things up. If any adventure gives the DM an excuse to poison the PCs' cups, drag their names through the mud, and send assassins to their inn rooms, it's an urban adventure.
Deadlines & Deadly Enemies. In an urban adventure, the PCs often have easy access to inns/lodging where they can long rest. Either you can create deadlines that put pressure on their time, or you can make fewer but much deadlier challenges. A deadline doesn't have to be "You have 3 days until the killer strikes again!" A good example of a softer deadline is "yeah, we can rest, but by that time the thieves' guild will have either cleared out to a new lair OR have fortified their lair with even greater traps/defenses."
What are the special challenges of an urban adventure?
The Law. You need to figure out what the law has to say about adventurers, weapon-possession and use, spellcasting in public, killing/apprehending without a warrant, the need for writs to adventure in certain areas of the city, and to what extent the PCs can expect support (or hindrance) from the city guard.
DMing Multiple NPCs. It's going to happen in an urban adventure. To be prepared, you want a list of names (organized however is suitable for your setting) & a list of roleplaying traits you can quickly access and check off. In actual play, you want to practice angling your body/head at different 45-degree angles to represent two different NPCs (plus voices/accents if you do that). Also make sure to keep track of the NPCs you create...because your players sure as heck won't remember.
How does the party's level affect the adventure? The higher level the party, the fewer NPCs in town who should pose a realistic challenge for them.
A CR 1/8 noble could very well be a challenge to a 12th level party. The PCs know she is bad to the bone, but don't want to run the risk of breaking the law until they get hard evidence of her alliance with the evil Dustwall orcs. She might have blackmail material on them, so that they don't dare risk moving directly against her until they destroy the evidence she has. Especially with urban adventures, it's helpful not to think in terms of the "encounter accounting" that's more common in dungeon adventures.
Dungeons in an urban adventure: yes or no?
Abso-frickin-lutely! I've run my fair share of thieves' guilds and wererat warrens beneath the streets. A great example was
Harbinger House for the Planescape setting, a multi-dimensional asylum.
Can anyone point me to some good urban adventures that I might study for inspiration?
I'm afraid besides
The Assassin's Knot and
Harbinger House, I haven't run any explicitly urban published adventures. You might find some suggestions in an old thread:
http://www.enworld.org/forum/showthread.php?56804-Best-urban-D-amp-D-adventure-new-or-old/page2