It is possible and is deceptively easy:
bin\Debug
folder below the project file (.csproj).app.publish
folder (they are not needed), and the .pdb files unless you foresee debugging directly on your user's system (for example, by remote control)), and provide it to the users.An added advantage is that, as a ClickOnce application, it does not require administrative privileges to run (if your application follows the normal guidelines for which folders to use for application data, etc.).
As for .NET, you can check for the minimum required version of .NET being installed (or at all) in the application (most users will already have it installed) and present a dialog with a link to the download page on the Microsoft website (or point to one of your pages that could redirect to the Microsoft page - this makes it more robust if the Microsoft URL change). As it is a small utility, you could target .NET 2.0 to reduce the probability of a user to have to install .NET.
It works. We use this method during development and test to avoid having to constantly uninstall and install the application and still being quite close to how the final application will run.