Not easily. You don't usually need the handle for gapidraw.net as the OpenDisplay functions can take the window title rather than the handle.
IIRC getting the handle is a pain in dot net. You have to do something like FindWindow (which probably requires p/invoke) to do it.
Unless you need the handle somewhere else, don't worry about it and use the title instead. That is how the GapiApplication class does it.
Sean