Most people don't have an umbrella handy unless they're using it to keep dry. And you don't need to learn martial arts (and it won't help you) to learn how to throw a chair.If you have enough time you could rip off a chair leg, if you don't have enough time you could just pick up the entire chair and use that. Same thing with umbrellas, if you don't have time to close it you could fight with it open or the umbrella could be closed in the first place in which case you won't have to close it. Usually umbrellas are kept closed except when you're using them to keep dry.
As for keeping the umbrella open, I don't know about where you live, but where I live when it rains it's normally windy. Try quickly taking an umbrella during rain/wind and using it to hit a friend as they mock attack you. It's difficult, and very unlike the stickfighting you learn in MA. At best the umbrella will act as a shield to block the first strike, which might be enough, but again not how stickfighting would teach you to use a stick.