I don't believe the Wolfram Cloud hijacks your right-click menus, as some other web apps do. This means that the right-click menu will default to what whichever browser your using would do. As graphs appear as images in the Cloud, the browser seems to treat them as any other image.
For example, in Firefox I can copy or save the image:
Generally I would recommend saving the object programmatically, for example using CloudExport which saves it in your Cloud directory. For example:
In[1]:= m = Graph[{1 <-> 2, 2 <-> 3, 3 <-> 1}];
In[2]:= CloudExport[m, "PNG", "myGraph"]
Out[2]= CloudObject["https://www.wolframcloud.com/objects/kmartin/myGraph"]
You can then share this object with the link (if you make it Public), or download it to your local machine.
Hope this helps, let me know if you have further questions.
