Filebeat is a great way to do that, yes. Be sure to indicate in the config that the data is already in JSON.
I personally don't know how to input data into an Azure cloud storage table. Logstash can write to a file. It can send data over plain TCP. It can send data via http. Logstash has many other output formats. If there's a way you can find to use these to send to an Azure cloud storage table, then great. Otherwise, you'll have to find your own way to extend Logstash, or read from a file, or listen to TCP and get the information over there yourself.
Are you referring to the metadata Elasticsearch writes out to the repository? Curator doesn't write that. Curator talks to Elasticsearch via API, and Elasticsearch does anything else. As with the other statements here, you'll have to figure out how to get that data "the last mile" into your storage table on your own.