Can I set aside the cake batter mixture and for how long?

It depends on the type of cake. If it's a sponge cake of cake with a lot of whipped egg whites it won't work very well but a traditional cake with baking soda or baking powder for leavening will work fine

I tried a similar thing when I was a student and the later cakes didn't rise very well. I am guessing that all the air I beat into the cake mix escaped standing their.

If I was you I would divide the recipe by 3 and make the cake batter in 3 batches you can make the next batch of batter whilst the first one is cooking.

It can be awkward dividing a recipe with eggs in, e.g. if it calls of 5 eggs dividing by 3 is a problem. in this case divide all the ingredients by 5 to work out how much per egg then, I would make a 2 egg batch per layer.

Hope that makes sense.

Yes, you can do that, with no problems. I've been baking for 45+ years, and have never had an issue with setting normal cake batter aside for later baking.

You shouldn't have too many issues doing that, I have before and no problems.