Any object placed into a geostationary orbit will remain there purely due to the virtue of gravitational and centrifugal forces. While external factors such as the gravity of other celestial bodies or the uneven mass distribution of the Earth will perturb the orbit, some light maneuvering thrusters will work just fine in the long term, provided you supply them with enough fuel. So the problem isn't keeping the city there, that happens almost by itself. Getting all that mass up there will be trickier.
Conventional rockets will work fine, as long as you have enough money and time. You can then construct the city directly in orbit, which shouldn't be a problem if you can get over the hurdle of launching that much mass up there in the first place. But there are better options than chemical propulsion available. Perhaps the best-known is the space elevator, a very long tether attached on one end to the Earth's surface and on the other end to a counterweight somewhere above geostationary orbit, kept erect by the centrifugal force generated by Earth's rotation. This dramatically reduces the cost of space launches, but it has its disadvantages. The material you're using has to be extremely strong, beyond any material we have available today. Additionally, it can only be built at (or at least near) the Equator, and there are issues with security and orbital debris, among others.
Another option is an orbital ring. This is a ring that spans around the Earth, kept from falling down by the angular momentum of a stream of orbiting material. I won't go into the specifics of how they work, but if you could build one of these, it would have tremendous benefits. An orbital ring can have sections that are stationary with respect to Earth's surface, and therefore you can build a tether from the ring to the ground relatively easily, requiring much less tensile strength than a space elevator would. Once you're in space, you can use a variety of techniques (perhaps something similar to a skyhook) to boost yourself up to geostationary orbit.
Incidentally, the ring would also provide a surface to build railways on (for ultra-high speed passenger/cargo transportation) or solar panels (without pesky weather to affect power generation). They're pretty spiffy. :p
There are many other options besides these, such as launch loops or space fountains. They are all grounded in hard science, more or less -- meaning they don't break the laws of physics, but they may require enormous feats of engineering and/or unplausibly strong materials.
One last note... all of this assumes that you're set on building your orbital city by lifting mass from the Earth's surface, but honestly, a better solution would probably be to mine a few asteroids for raw materials and use that instead. Depending on the specifics of your world, this option might prove much cheaper.