By default, your AppServices run in UTC. There is an almost undocumented (or at least very little known) feature of KUDU which allows you forcing a specific time zone to your application. By adding specific Application Settings to your App Service (in Azure Portal) you can set following features:
WEBSITE_TIME_ZONE, e.g.Central Europe Standard Time(List of supported values)SCM_TOUCH_WEBCONFIG_AFTER_DEPLOYMENT= 1/0 – does what it says (default is 1)SCM_TRACE_LEVEL= 1..4 – you can get more detailed tracing by setting higher level (default is 1)- Build/GIT related options
- Caching related options
- Setting Node/NPN version
WEBSITE_LOAD_CERTIFICATES– Loading certificates- Low-level diagnostic switches
WEBSITE_DISABLE_SCM_SEPARATION– allows you to run your site and SCM (Kudu) in the same sandbox (and potentially save some resources) – unfortunately marked as legacy-obsolete and not supported any more- and many more
You can find complete list of features in Project KUDU Wiki.
HAVIT Knowledge Base 
Can the WEBSITE_TIME_ZONE be added to web.config/appSettings? Seems like no.
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No, unfortunately not, as it is interpreted by KUDU, not your app.
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