Keeping a cake moist starts with making a moist cake. Make sure you do not overcook the cake or it will be dry non matter what you do to it afterward. Frosting a cake is one of the best ways to lock in moisture, but even the frosting must be protected from becoming brittle. Covering a cake properly so that it is both protected from air and from destruction requires a little creativity and caution to make sure it is as good the next day as if it was just baked.
Frost before going to bed. The best way to keep a cake moist overnight is to cover it with a moisture-locking frosting or icing. The icing acts as a shield against the air and allows the cake inside to maintain its moisture. So if your cake is going to be frosted, don't wait until the following morning unless your frosting is time sensitive (such as a whip cream frosting).
Place the cake on a sturdy plate and cover with plastic wrap if possible. If the cake is frosted or sensitively decorated, plastic wrap is not a good idea as it will smear or flatten the frosting. Plain or fondant-icing covered cakes only need tight fitting plastic wrap to stay fresh overnight. Store in a cool dry area.
Place the frosted cake on a large plate or tray and invert a large bowl, baking pan or pot over the cake if it is the correct size. This will protect the cake from air without ruining the frosting. If the cake is too big for any invertible item you own, find a large cardboard box that would fit over it and line it with wax paper. Though it is not as airtight as a plastic or metal bowl, it will keep the cake moist overnight.
Cover and place perishable cakes such as cheesecakes, tiramisu and cakes that contain cream fillings in the refrigerator. Though refrigerators are not ideal for cakes as they can cause excessive moisture, they will keep them safe if properly wrapped or covered. Remember to never put a fondant covered cake in the refrigerator as the fondant will sweat and melt once it is removed.
Line a large plastic locking container or lidded cardboard box with wax paper and place as many cupcakes as will fit inside. It is best to put cupcakes inside of closing containers as they tend to dry out faster than cakes due to their size. Make sure the cupcakes have enough space between them that they don't touch and that the lid clears their height.