You can indeed use an umbrella as a weapon for self-defense. They even sell metal ones for precisely that purpose.Real Self-Defense: Unbreakable Umbrella Under the video section on the website, you can find some instructional videos that may give you a few ideas. However, it’s important to remember that a normal umbrella will work differently from one that has been specifically designed around combat.
You can use it several different ways (that I am not an expert on) including using the end to poke someone and opening it in an attacker’s face. Most the techniques you can use with an umbrella are similar to the ones you can use with a walking stick, a short stick, or another long implement such as a tire iron or a poker found by a fireplace. The best thing to do, usually as an untrained fighter, is to reinforce yourself, hold it with two hands and stab at someone toward the midsection/chest area to knock them backwards.
Remember, the goal of a character who is protecting themselves isn’t to win the fight, it’s to do enough damage to the other person so that they can escape. This is something that’s commonly forgotten by writers because the sequence where a character stands up for themselves is usually tied to them coming into their own. It is very powerful, but is sometimes used so often (and in ways that make no sense) that it can get a little eye roll worthy. Again, self-defense isn’t about standing over the aggressor so that they know they’ve been defeated, it’s about escaping from a bad situation.
Almost any item can be used as a weapon, from a plate, to a coffee cup, to a can of beans, to a bottle of Jack Daniels (Punisher #Dirty Laundry WARNING GRAPHIC VIOLENCE), and others. Here’s a video from Michael Janich where he goes over some techniques using Improvised Weapons.
Remember, though, a weapon is only useful to your character if she practices with it. As writers, we sometimes have the common failing of assuming that the weapon is the important thing. It’s not, the wielder and what the wielder knows is what’s important. A weapon is just a tool and even the most unlikely items can be great weapons in the hands of a character who knows how to improvise.
Also, the Amelia Peabody books by Elizabeth Peters showcase a character who uses an umbrella for self-defense. You should definitely check those out.