Sorry, I don’t recall in my decade or more of using Notepad++ ever seeing an always-open-in-administrator-mode right-click option.

If you want that to always be the case, you might be able to use Windows OS techniques to accomplish it: right click on the Notepad++.exe executable, and go to the Compatibility tab, and select the appropriate checkbox on that tab. I think that will make it so Notepad++ will always open in Admin mode, even called from a shortcut or a right-click context “open file”/“open file with” action. But Admin mode is controlled by the OS, not by the application, so the final word has to come from the OS.