Now-a-days, everyone is heading North. If you are planning a "all-boys" trip, go ahead, as Jojikhan said. But if you are travelling with family then better wait for a few week for Bhair-chaal to calm down and cost/charges to be reasonable.
I also have City 2016 (provided by government) and its road clearance is "wahyat" with respect to Pakistani road conditions, especially in hilly areas. Now I have raised it by 1'' Aluminum spacers on all 4 wheels. Its better now. Another possible improvement (that I cannot do on my car because of govt rules ) is replacing low height factory fitted tires to new higher tires. Any tire dealer will adjust the price of original tires and give you new tires at a reduced cost. You can gain another inch of road clearance by this method. Thus your city will be better prepared for North. However, even then, be prepared for scratches, bruises, bumper touch and under body hitting at certain locations (like a partially cleared landslide).
Take babusar top route, if you are first timer to North, be extra careful especially at down-hill steeps, keep the car in low gears and low speed and avoid frequent use of brakes. Avoid a complete halt at uphill road, it will be difficult for you to move again and even worse for the traffic behind you.
Honda city(may be other new cars also) oil chamber is made of Aluminum alloy and it is the lowest point in Engine. It is prone to crack if hit by a stone or stone-like thing. Go under the engine and see this by your self for a better idea to avoid a hitting from beneath.
As per latest info by Respected Jojikhan, road up to Khunjrab is "Qabil-e-Rashk" so no problem with City. (revised on 12-7-16)