That stinks. My Ninja is 4 years old, so I'm guessing it isn't compatible with the newer sockets, but I'll try to figure it out.
In these forum there are already some indications about how to mount a Ninja on 1156/1155/1150 sockets: check them out.
how much I'll have to spend to get something that will work well with a quiet Nexus 120mm.
Providing you won't overclock, a modest Cooler Master Hyper 212 EVO should be enough (but there are also alternatives from Scythe and Thermalright).
I don't know whether a 3-pin Nexus would work well with current 4-pin PWM CPU headers.