“We can start this night”
Normal use should surely be “tonight” not “this night”?
In most cases, esta noche = tonight or this evening. On the other hand, you might say "this night" if you were pointing to a specific date on the calendar. https://image.shutterstock.com/z/stock-photo-finger-pointing-into-calendar-planning-concept-167243117.jpg
Sure it's possible, but usual use is in the sense of"tonight". Why make people search for a context? Or why not provide one?
I've already agree with you -and- I've provided one possible context for "this night." So, I'm not sure to whom you intended your reply.
Why did you originally ask this question? Where did you see it translated as "this night"?
It sounds OK to me, depending on context, but I suspect there are grammatical traps here for non-native speakers. To be more emphatic you could even say "We can start this very night".