The Coolpix WILL take much better photos at night than your phone. To be fair, just about any cheap compact will give pretty similar nighttime shots. The limiting factor is the small sensor.
You are going to have to spend quite a bit to make a significant difference. You can increase the quality of night photos a bit by learning how to set the ISO of the camera. Don't let it go above ISO 800, even for nighttime shots.
Also, if you are taking pictures of buildings, learn how to turn the flash off as the flash won't be strong enough to light the whole building. Without flash and with low ISO, you will need a small tripod or POD to stabilise the camera. I recommend you try the camera out at night in your town ASAP and see if you can get the sort of photos you want.
If you use a tripod and use one of the night scene modes, it will result in a better photo than a smartphone will. They might not be as good as a DSLR, but they should at least be acceptable. However, using a tripod and night scene mode are both essential.
You may want to search You Tube. I think there are some tutorials on taking night photos using compact cameras. They probably won't be specific to your camera, but the info should be general enough that you could apply the techniques to your camera.
However, practice ahead of time as it might be tricky to get it right - especially if there are a lot of street lighting in your scene. You don't want to wait until you get to Times Square to experiment.
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