IntelliJ shortcut to show a popup of methods in a class that can be searched

I prefer to use the Structure view. To open it, use the menu: View/Tools Window/Structure.
The hotkey on Windows is Alt+7, ⌘+7 on OS X

Use Navigate (View in older versions) | File Structure Popup (Ctrl+F12 on Windows,
⌘+F12 on OS X). Start typing method/symbol name to either narrow down the list or highlight the desired element. Press Enter to navigate to the selected element.