On the old Apple branded external keyboards, the menu button is to the right of the space bar (Alt Gr, on PC style keyboards). Looking at the newer keyboards, and the Mac Book Pros, it doesn't seem to be there (guess it went the same way as the Home, End, Page Up, Page Down and Delete keys).
Having a poke around the Preferences -> Keys preference page, I don't think there is a way of revealing the entire context menu.
Many of the sub-menus are available from the keyboard - the ones I have committed to muscular memory are:
- Alt ⌘ T - refactoring menu, and the various refactorings available from the keyboard, using the modifiers Alt ⌘
- Alt ⌘ S - Source menu, and various operations beginning with Shift ⌘ - including organize imports, reformat.
- Shift ⌘ T - open type
- Shift ⌘ R - open resource
If all else fails, it is usually worth looking at Shift ⌘ L - which shows all current keyboard mappings in a hover on the side of the screen.
With these shortcuts, I would recommend learning one a day. Their culmulative effect is considerable.
I appreciate that this does not answer your question, but I hope it solves your problem.