In the examples you make, there is no need to use any preposition.

> We can get together Wednesday night.  
> Since there was nothing I could do, I wandered in downtown the night of Christmas Eve.

At least in American English, you can use a weekday as adverb, such as in the following sentences:

> We will try again Friday. (We will try again on Friday.)  
> See you Wednesday. (See you on Wednesday.)

This usage is reported to be chiefly North American, from the OED.