How do I access Android STK Menu using APK?
This is, unfortunately, not possible. SimToolkit applications are protected from use by other applications because they could incur charges for the user or alter his connectivity state without his knowledge or consent.
The SimToolkit application (which is signed by the manufacturer or mobile operator so that it has special privileges) is the only application which is permitted to communicate with the STK application on the SIM card.
The STK as in Sim-Tool-Kit, is actually available in the source code for AOSP.
It can be found under packages/apps/Stk, so it would be a matter of taking the source from that and modify to suit your needs.
I downloaded the stk app from here, https://goo.im/devs/playfulgod/misc/Stk.apk/ and copied it to system/app , modified the permissions to rw-r--r-- and rebooted. Voila! Stk appeared on the app drawer and it's working. Using Paranoid android 4.5-Beta2