How to execute update ($set) queries in MongoDB Compass tool?

You can use mongo MongoSH. Currently its in Beta.

In the latest version, there is a "_MongoSH" in the bottom left corner of the window. Thx to @Boštjan Pišler for the hint about a new feature.

Old answer: I had the same issue, it looks like a simple feature to implement (since document updates are possible) but... AFAIK there is no such option in compass, you can do it through mongodb shell (CLI client).

The tool was recently updated with a shell (currently in beta) which is accessible by clicking the dark bar on the bottom of the window "_MongoSH"