Did the bracket have screw holes in it? I ask, because most of the things have double-stick tape on them AND screw holes. The tape is to hold it in place so you can get the screws installed without it moving around. The tape does NOT hold it permanently.
Now, there are some permanent double-stick tapes out there - often used in the auto industry. Still, you may want to use screws. The surface must be very clean to use the tape - any pre-existing soap film must come off first if you want it to last.
Acrylic should be fairly easy to drill. Depending on how thick it is will determine the type of fastener.
You could also try using silicon. Make sure that you don't put any pressure on it (tape it in place after smoothing/cleaning the edges) until it has a chance to fully cure. Depending on what you use, this can take anywhere up to several days.
There are hotmelt polyurethane glue guns that would probably work as well. This gives a good bond in less than 90-seconds, but they are not cheap (a kit costs about $100). The glue also has a shelf-life, so you can't just leave it sitting around until you need it. Don't confuse this with a more common hot-melt glue using sticks. Those soften again when heated, the polyurethane ones melt once then chemically cure - they will soften a little, but never reflow like the typical hot-melt stuff can.
Now, all said and done, I've not tried this myself. I have used, and own, a Hi-Purformer brand ployurethane glue gun, and for those things thatcan use it, it is really handy.