If you don't want screw holes showing on the top of the box, get some threaded nylon standoffs and glue them on the inside of the box. Glue will depend on what two surfaces you are gluing to. You will have to cut the standoffs to the height you need.
For accessing the switches, you could drill holes as you say, but the buttons will not raise above the surface of the panel. There are different ways to deal with that. One would be to use rubber feet like these (we can find these in hardware stores) https://www.sparkfun.com/products/10594
I tend to use these type of switches when I do something like this:https://www.sparkfun.com/products/10302
If you have them available locally.
One tip I have for when cutting your panel (or even drilling) is to place a few layers of painters tape (masking tape) around the area you will be cutting. This helps protect the rest of the panel from slips. Also, it allows you to draw your dimensions right on the tape to help you when you cut. When you are done, you just peel the tape off. Also cut from the back side, not the side to be seen.
For rectangular cuts, drill holes along the inside of the rectangle edge. This will make it easier to cut. If the material is thin like crossroads posted, you can just use a razor knife to finish it. If it is thicker a side-cutting bit in a rotary tool might work.