No, this will not work, your Z170 mainboard does not support this processor. You can only install 6th and 7th generation Intel processors on your mainboard, 8th generation and newer need a different board. Conclusion: In addition to this processor, you need at least one new mainboard with suitable RAM.
I use the Corsair AIO iCUE H150i Elite cooling with this processor. I have set the TDP from 65 to 125 watts permanently, yet the processor only gets about 72 degrees warm under permanent full load. Works great.
I run it with a NZXT Kraken AIO water cooling system. I would recommend the Noctua NH-D15 air cooler, but be careful whether it fits on the mainboard and in the case. You can read about this at Noctua.
There is a fan with it. But it is not worth anything. CPU gets too hot. Normal operation 90 degrees Celsius. With large CPU cooler approx. 50 degrees Celsius.
The linked cooler should definitely be sufficient, the i7-13700 is significantly less power hungry compared to the K models, and therefore easier to cool.
According to https://www.asus.com/motherboar..., this should fit together, but only from BIOS version 2004. Thus, a BIOS update might be necessary depending on the situation.