To maintain animals, you need plants. To maintain plants, you need warmth, light, or ideally both.
Your demons and monsters are unlikely to take a side salad with their meals, so they would probably prey on the largest non-monstrous creature in the area. If a monster eats like a carnivorous human, let's say each monster needs 50 pigs a year, one per week, ish. Supporting large animals like that requires some serious calories to be available.
To this end, I would suggest making your city home to geothermal power, which fuels the growth of dark-leafed plants , mosses, and lichens that survived 99% of the lights going out. Maybe that's why that location was chosen for the city in the first place? It's not unreasonable to think that these near-future people could gene-splice together crops tailored to grow underground; imagine a field of wheat, or an apple tree, almost jet black. Over a thousand years of neglect, basically everything broke down, including the heat transfer conduits, meaning a large number of physically disparate locations within are overrun with plant and animal life, all supported by heat not going where it was designed to go. And as a bonus, this provides light in some places, in the form of dully glowing metal.
So, with all that out of the way: vehicles would be rusted to unrecognisable scrap, and completely devoid of paint. Ditto signage. 121
Windows would probably be alright, unless the heat's really something to write home about in some places, in which case they would likely have broken from either sustained heating or repeated heating and cooling, depending on how static the environment is. Note that suspended windows, like the wall panels of large public buildings such as art galleries, are held in place using metal fasteners that will definitely have given way. Electronic circuitry is a non-starter, as others have said, but it's always possible to fix some stuff with Handwavium technology.
Buildings are a little more tricky. Some concrete mixtures will last centuries, some will be crumbling in fifteen years. Some buildings don't even use concrete anymore, or if they do, only in the foundations. An underground city doesn't sound like something that would spring up organically, which implies it was a designed habitat, which implies that it would have been put together all around the same time, or at least sufficiently close enough together that after 1000 years it makes no difference. The key, then, is consistency: decide how unstable and crumbled your buildings are, and stick to it (at least by category; steel-framed buildings would definitely have collapsed, long ago, and rusted into almost nothing).
Personally, I'd go a little unusual and flatten your city a bit, make it so the wildlife and passage of time have made it into a semi-open plain, with concrete pebbles and dead things forming a semi-toxic soil underfoot, dotted with outcroppings in the form of old buildings. It's not quite a maze, it's too open for that, but it's certainly not somewhere your average line of sight is very long. If you want to gain access to any of the old buildings, you'll need to bring seismic imaging equipment and a shovel. 111