I have a Website, which is Fully Ajax-Based (Hash Navigation).
(When I Click on a link, the Tags, and there values should Change)
No. Open Graph markup must be present on HTML pages which are GETable with pure HTTP.
This is because when a user interacts with an OG object (like, performs an action etc) Facebook will perform an HTTP GET on the OG URL, and expect to see OG tags returned in the markup.
The solution is to create canonical URLs for each of your objects. These URLs contain basic HTML markup including OG tags.
On requests to these URLs, if you see the incoming useragent string containing 'facebookexternalhit' then you render the HTML. If you don't, you serve a 302 which redirects to your ajax URL. On the ajax URLs, your like buttons and any OG actions you publish should point to the canonical URL object
As a user, I'm on http://yoursite.com/#!/artists/monet. I click a like button, or publish an action, but the like button's href parameter, or the URL of the object when you post the action should be a web hittable canonical URL for the object - in this case, perhaps http://yoursite.com/artists/monet
When a user using a browser hits http://yoursite.com/artists/monet you should redirect them to http://yoursite.com/#!/artists/monet, but if the incoming useragent says it is Facebook's scraper, you just return markup which represents the artist Monet.
For real world examples, see Deezer, Rdio and Mog who all use this design pattern.