Well, though this isn’t a startling discovery, I found out you can start up any of the GUI apps just by running them from a commandline while ssh’d into the iPhone. Just follow these instructions to install ssh, then run bash. With bash running, just cd to a directory:
# cd /Applications/Calculator.app
and run the program:
There is no user interaction available, I’m afraid. Then again, perhaps there is*… In order to quit the app, you have to shut down the ssh session.
*pure wild speculation, I have no connections inside Apple : (