I can tell you how to add commands to that context menu: under the Tools menu -> Customize, in the Commands tab, click "Context menu:", find "Project and Solution Context Menus | Solution" in the extremely long and unwieldy dropdown, then hit Add Command... and add the command you want in that menu.
I've only used that information to add a macro (to replace the native "Open Containing Folder" command with one that used a different file manager), I've never tried to create my own command that wasn't a macro, but I expect if you were able to get custom Commands in custom buttons, you might also be able to find them in the Add Command dialog as well?