Build one web project from a Solution with multiple web projects in Visual Studio Online Build Definition

Instead of choosing the .sln file, you need to select the .proj file.

Firstly, click the ellipsis button next to the Solution field.

Then choose the .proj file you want to build. For example, the screenshot below shows that the build definition is configured to build the ClassLibrary1 project, instead of the whole solution (contains ClassLibrary1 and ClassLibrary4 project)

You can add a new "Azure App Service Deploy" task to your build definition and set the "Package or folder field" to "$(build.artifactstagingdirectory)**\YOURPROJECT.zip"