So like, you can have the compass mounted on the inside of the "dome" (forgive me if my umbrella terminology is lacking). So when you're walking around outside and it's raining, you won't get lost for not being able to see the sun. When the sun finally comes out, just turn the umbrella upside down and
point a certain spot on it to the north, and the shadow of the specially designed stem will tell you what time it is! Collapses and folds for easy storage, naturally.
P.S. I was actually thinking of making the "dome" out of clear plastic, have compass markers on it, and allow it to spin freely on the stem, some of the spokes being magnetized, perhaps. I just figured wind would probably make it useless then, so a compass mounted inside somewhere was the next best idea.