If you want to update your website, you can use FTP,Git or other tools. You can refer to this document which can tell you how to deploy your sites. And contains other way to deploy your app. According your describe, you can check the last modified time of files in kudu.The following paragraph will introduce kudu.
If you just want to update your static files in sites. The easiest way is logging on portal and find your app.
First, you should click Advanced Tool in left side, and click the link to open KUDU Management.

Second, find Debug console options and click CMD or PowerShell, then you can see folders in the page, click site->wwwroot .

Third, you can add/delete/modify your static files by buttons in the page. When click the icon of pen, you can modify this file.

If you want to copy your local files, you also can drop it to this page, it's very convenient. More function about Kudu you can see this site.
Hope my answer works for you.