Many plastics work with the same tools used for wood. Plastics are abrasive and dull tools quickly, though. The problem with acrylic is that it likes to crack, especially when you are drilling near an edge. We always used the same drills we used for wood, backed the material being bored, and had extra blanks. When I picture how I made my router base, my suggestion is to just cut an opening in your base with a coping saw, perhaps backing it with some scrap material if it is particularly thin. Just cut in from the edge to make a suitable opening, no drilling required.
By the way, the tang of a file (where you’d mount a handle) makes a decent deburring tool for plastic if the edges of the tang are crisp. Use the edge of the tang to scrape the burr off the plastic. A piece of scrap soft steel is even better but you’re more likely to have a file sitting about.