I am trying to find if that is possible to change site's search navigation(a.k.a search dropdown links) through CSOM?
Seems Search's export configuration does not contain info about search navigation(Site Settings->Search->Search Settings), and i am struggling to find any article that describes whether this is doable through CSOM.
In server-side code, you'd use the SPNavigation.SearchNav property to customize the search menu. However, the client-side object model doesn't include that property. Instead, you have to access a special "search root" node as documented in the blog post I linked to and add/remove child nodes to/from that root node to customize the menu.
I used that approach recently to write a scheduled job in Windows Azure that copies search navigation settings from a parent site to all the child sites in my collection (since SharePoint 2013 doesn't let child sites inherit search navigation settings, unfortunately). Since the job was hosted in Azure, I had to use the client object model to accomplish this.