Cities here on earth already have a wide range of problems with weather, the city-planet would have to deal with them, too.
Water
Cities have drainage problems. There's not enough soil for all the rain to sink down in, and the city will either often be flooded, or it'll have to have some serious drainage channals. If your planet doesn't have oceans, it'll have to dot cloud factories around the planet to get it to rain. And you'll want rain, because the rain will clean the air just a bit. This all has to be included in the urban planning of the city. If the contractors didn't include heavy-duty drainage back when some section was built the streets will turn into a swamp.
Wind
Again, if you don't have oceans, you can expect your wind to be a bit like on venus; from wikipeda:
Thermal inertia and the transfer of heat by winds in the lower atmosphere mean that the temperature of the Venusian surface does not vary significantly between the night and day sides, despite the planet's extremely slow rotation. Winds at the surface are slow, moving at a few kilometres per hour, [...]
The wind would be slow and warm, heated by the sun and the city itself. It would not often change directions or do anything interesting like spin into tornadoes, this is because the high and low pressure areas are basically only determined by what the sun heats. Forget about wind turbines to generate power. And all that is at highrise level, on the streets, there will be virtually no wind.
Your cloud factories are of course uniformly distributed because no section of the planet wants to live without rain, and as such, these will do nothing to cause the pressure differentials needed to whip up the winds.
If you have problems with smog on your planet, they'll be worsened by these slow winds, your smog will just hang over your factories and highways without dispersing, and only the rain will eliviate this a bit.
One advantage of these slower winds is that you can build higher, since there's less lateral stress on your buildings.
With oceans things are better: You could expect more earth-like winds, at least high over the streets. The stale air down below the highrise would not be helped much.
Sun
If you still have problems with smog, then the answer here will be clear, it'll just be hot and filthy during the day, and it'll be warm and filthy during the night. In case you managed to keep your skies somewhat clear, things get interesting.
Anywhere with highrise, the streets will be cold, only heated by the activity going on there, but at least there's no wind. The rays of light that make it through at noon or that bounce off of the glass faces of the towers are far too short-lived to heat the place up.
The higher you get, the more light you will catch and the more pleasant it will become (hey it's a bit like being a tree in a forest), there's sunlight and at least some wind too. Yes, the most prized real-estate is definitely high above the city streets, so that's where you'll want to build, higher and higher.
You could maybe line all your buildings with mirroring glass to get the sunlight down, it would be expensive and confusing and cold inside the buildings but at least you might have a chance of heating the streets.
Waste
There's energy in waste, and if you won't take it, nature will. Waste disposal is problematic in cities here on earth too, endless piles of garbage in the bad parts of town, problems with rats or badgers. Your planet will have these too, in its own form. If your city-planet has been a city for a couple thousand years, you might get very specialised flora and fauna that'll help clean up the forsaken streets. The people will hate the sight of them, but they make things better, really. Try to live with them instead of fighting them, maybe employ them, scatter their seeds and set them loose on filthy streets. Then when the waste has degraded, scoop up the leftovers and hose it down so people can live there again.
Some form of standardised and automated garbage disposal ought to be built into every building for the more respectable parts of the city. When your population gets dense enough there is just too much waste being produced for man to keep up with, so your options are to either automate it (I can't imagine that the entire planet was built from the ground up to support this, and with it being an afterthought, many districts are without), or nature will take over (not great if you want a planet-wide city). Garbage, an ever-growing problem on your planet, will be your biggest issue and I think a very large part of your population will be concerned with this, including politicians, engineers and even merchants as your garbage may make an interplanetary export product.
