Sewers still work the way they always did
The DWV (drain-waste-vent) system in a high-rise building is actually quite similar to that found in a house -- it's simply bigger! The principles (gravity flow, mostly) that make a building's DWV system work scale quite beautifully with the size of the building.
As to the sewers under the streets, you'll have separated sewers for sure in a city like this -- you simply have too much sanitary flow to afford spilling raw sewage due to rain-induced sewer overflow events. Your buildings will have roof drain systems that simply tie into downpipes to the storm drain system, while their DWV system ties into the sanitary sewers. Both of these systems will be underground rivers, basically -- at its most extreme, you get a "river atop a river" effect, with the storm drain system built atop the sanitary sewer system, and your streets built over the storm drains, with utility ducts/tunnels flanking them for other utilities.
Pump up the pressure to get water to go where you want
In order to get water to the tops of such high buildings, high-performance pumps are mandatory, both for fire suppression and domestic service. These superbuildings will have redundant fire pumps (due to the criticality of the service involved) feeding combination wet head sprinkler/wet standpipe systems (there is no other way to control a fire in an Earth-sized high-rise, never mind a Coruscant-sized one -- One Meridian Plaza and First Interstate Bank taught us that already), in addition to a pump dedicated to providing enough domestic water pressure to reach the top of the building. Pressure-reducing valves will be needed on each floor to prevent the high standpipe/vertical main pressures from reaching fixtures and damaging them, while the pumps themselves will need to achieve pressures likely into the thousands of PSI (such pressures are more typical for motive hydraulics than water service), and you will need break tanks every so often to keep the pump pressures from becoming totally unreasonable. A good break tank setup would pump up to the tank and then gravity feed down, by the way -- this way, redundant pumps can be used easily, and even if all the pumps fail, a limited supply of water will be available.
Evening out hot and cold
The square-cube law works in our favor here -- enlarging a building is beneficial from a thermal performance standpoint as the marginal heat loss goes down as you further increase size. However, the heating and cooling loads are going to be high, still. A high-performance, lightly glazed envelope (vs. the glazing-everywhere postmodern high-rise aesthetic) is going to be a necessity in these superbuildings, and they will likely be forced to rely on distributed ventilation in order to allow the structural design to prevent stack effects by placing air barriers between floors, with either mechanical floors feeding transfer media (steam, water, refrigerant) to air handlers in each compartment, or complete HVAC on a per-compartment basis. Domestic water heating will be handled the same way -- either by indirect tanks off the HVAC heat loops, or by per-compartment hot water heaters.
Big power means big problems
Last but not least, we have the electrical and communications infrastructure needed in such a megabuilding -- a set of medium voltage "trunks" with accompanying fiber optics will be run in heavily firestopped vertical shafts in the building core along with the other building services, with dry-type transformers on each floor to provide low voltages for lighting, receptacles, and appliances. Said transformers will likely be tied together in what's called a secondary network to bolster the reliability of the electrical service, while the fiber optics will feed distribution nodes on each floor (similar to a cable-TV hybrid fiber coaxial node, or a passive optical network splitter for that matter, with primary voice service being provided via some type of Voice over IP setup).
Redundant trunking will be provided for both power and communications in order to prevent a single failure from knocking out power or voice/data services to the whole building, while secondary systems may also be present for power (such as a low voltage auxiliary communications power system to play the role of the 48V central office batteries in a POTS system) and communications (a firefighter's telephone system will be needed as handheld radios are no good for a fire crew working in a highrise). In addition, key services (fire/life safety) will have their own backups for power and the likes (including dedicated generators or engine drives).
All of this discussion ignores one utility that is necessary for every high-rise, though, and that's egress. Your superbuildings will need a high-capacity, high-performance egress system in order to get people out safely in case a fire, chemical release, or other emergency makes part of the building untenable. While total evacuation isn't generally required in high-rises thanks to modern sprinklering and fire-resistive construction, egress in a Coruscant-sized building becomes much more challenging. In addition to the pressurized, smokeproof tower stairshafts used in today's high-rise design, such a megabuilding will likely need to use smoke-protected occupant evacuation elevators in order to get people out from a "safe" floor one or two floors away from the emergency situation to a designated point of refuge or further exit means, as well as using any skybridges present as horizontal exits so that evacuees can seek refuge in neighboring buildings.
Exit capacity will be another issue as well. Occupant evacuation elevators, while providing accessibility and a degree of speed over long vertical distances, struggle with moving large crowds in a smoothly flowing fashion. Stairs provide high capacity and smooth flow at the cost of space and physical fitness requirements; even then, though, you will likely need more than a typical high-rise core's set of stairwells to provide for full capacity egress, especially if high-density occupancies such as assembly halls are present on upper floors. What are called scissor stairs can provide a useful boost to capacity per unit volume of space, although they have the downside that whatever renders one uninhabitable will likely render the other stair in the scissor-stairs uninhabitable as well.
And that's before you get into the challenges that pressurization zones would add to this all! Putting airlocks in egress paths is a major obstacle to mass egress, never mind pressure differences acting on exit doors to the point where without careful engineering, exits may become utterly inoperable.