How can I run a custom .sh script at each startup (without native init.d support)?
I think that most easy way would be by using app Kernel Adiutor. It has ability to emulate init.d
Open app and swipe from left edge, it will reveal menu, scroll to tools section and than tap on init.d 
There you have a switch to enable or disable init.d emulator, for devices that doesn't support init.d, in bottom right you have a "plus" icon to add scripts to run.