I recently upgraded from Lucence search to Azure search for one of our client and yes 1000 field limit very annoying thing with Azure search because the default Sitecore installation has that much of fields.
So you can choose two way to handle this -
1.You can set the following setting to false
<indexAllFields>true</indexAllFields> in Sitecore.ContentSearch.Azure.DefaultIndexConfiguration.config and manually add the fields those are required in your search in Sitecore.ContentSearch.Azure.Index.Web.ExcludeFields.config
Or you can add the template id's that is required in your search instead of a particular field.
Or you can both step1 and step2 together.
Or yon can go with another way by updating the Root of index in Sitecore.ContentSearch.Azure.Index.Web.config
<crawler type="Sitecore.ContentSearch.SitecoreItemCrawler, Sitecore.ContentSearch">
And if you use the above configuration then content editor search will also work.