Translation:On your big evening you can decide everything.
"An" (at) is a preposition that can occur with both dative and accusative. When it is used to describe a space, it is fairly easy. A movement that stays within the given space is described with dative a movement into the space with accusative. Here, "an" is used for time, where English would use "on". Again, dative describes an action within the given time frame. (The accusative is not really used with an for time. You would rather see "bis".) Here, the evening sets the time frame for the complete action, so "an" and dative are used.
Try this for an in depth reflection on the reflexive.
Well, I think that for your big night would be more like something you are going to do FOR someone on their big night as opposed to doing something ON their big night.
For your birthday, [I gave you $100], [we went to the movies], [I will do this or that] ... To me, in English this doesn't specify a definite time. The action was just done 'for' the birthday .
On your birthday, [we will walk], [I will give you $100], [I want to go swimming] ..... Clearly, every action mentioned here will be done ON the day of the birthday..
We might say "In the evening of your big day you can decide everything" but "You can decide everything on your big night" may be a bit closer to the intended meaning. We say "your big day" and "your big night" but I've never heard "your big evening" or "your big morning".