Making batter one day, baking the next
It most likely will not work well to do that. Unfortunately, baking powder (in this case in your self-rising flour) starts the chemical reaction that gives cakes their lift as soon as the powder meets liquid. If the baking powder is single action, all of the bubbles are created when the water meets liquid and the heat of baking doesn't really play a role. If the baking powder is double action, some of the leavening reaction occurs when the batter is mixed, the rest of it happens while baking. So, if your self rising flour is made with single action baking soda, the cupcakes will fail if the batter is held overnight. If your self rising flour is made with double action baking powder the cupcakes will not have all of the lift they should, but may be OK. I have many times saved pancake batter for 24 hours and used it with no problem (flour, double action baking powder, baking soda, salt, buttermilk, eggs), but optimal lift isn't as crucial in a breakfast pancake as it would be for cupcakes.

That thought does present a possible "out of the box" solution to your problem. Could you cook your leftover batter on the stove while the cake is baking? I would try one in a skillet with a lot of butter, flip it when it's good and bubbly. I don't know where you are from or if pancakes are something with which you are familiar, but pancake batter and cake batter are very similar. They wouldn't be cupcakes, they would be something else entirely, but they could be very good.

If you decide to make cupcakes with day-old batter and your self rising flour is made with single action baking soda, or you're not sure, you should make your own "self rising flour" to give yourself the best odds. For 2 cups (`225 grams) of cake or pastry flour (soft, lower protein than all purpose) add 3 tsp double action baking powder and 1/2 tsp salt.

I'm assuming that you've considered cooking the cake and the cupcakes simultaneously in the oven and you have reason to not do that.

(Edited after Joe's comment about single vs double action baking powder)