We currently host a couple of nop sites (3.90) in azure (webapps). The repos are in VSTS and bitbucket, built in VSTS and deployed to azure using the VSTS deployment features. we're not using any caching so i cannot comment on that, but otherwise it works fine. If you inspect the web.config you'll see some azure related config settings.
If it is not a secret, could you please share your build configuration in VSTS? It would be really helpful!
I was trying to set it up but still, have some problems with generated artifacts.