Easiest option is if you know you will be hitting a wall stud (use a stud finder or hammer a really thin nail into the wall and if you feel resistance after you go through the drywall, then you know there is a wall stud there). Just screw through the display case backing, through the drywall and into the stud. Done.
If there is no wall stud where you need it then you would need to use wall anchors. If you don't, then over time, the screw will just get pulled out of the wall. Use the supplied mount and hang your frame with that. While it is hung, screw through the backing of the case with a really thin drill bit. This will mark the case and the wall with the location of the screw/anchor hole. Remove the display case from the wall. Now use an appropriate size drill bit (slightly smaller diameter than the wall anchor) and drill through the drywall at the spot that was previously drilled with the thinner frill bit previously. Now tap your anchor into the wall with a hammer, rehang you display case and screw through the backing where you already drilled through and secure to the wall and into the anchor.