No, you can't pass arguments from the Start Menu shortcut. If you find a way, let me know because I've been trying to do it for a long time :).
Here's how I address the problem in my situation...
If the ActivationUri property is not null, the application was launched from a browser and passed arguments via the query string (just like you described).
If the ActivationUri is null, the application was launched from the start menu. In these cases, I get the arguments from a config file deployed with the application.
What type of information is being passed with your argument? That would be useful in order to determine how you should store your arguments (app.config, application settings, user settings, isolated storage, etc.)
Side Note
I thought I was on to something by making my application "single instance" and trying to handle the StartupNextInstance application event, but no luck. You can only get traditional command line parameters that way, not the ClickOnce query string parameters.