Hi, is it possible to get my nat ip address across the Internet?
By normal means no. Your internal address will be mapped to your router public address and the information lost to anyone outside. When they connect back to you (being an active connection or a response packet from a connection you established), your router will remap the public address back to your internal IP address (making also any necessary changes to the packets contents) without the outside knowledge.
However, if someone from the outside somehow gains access to your router translation tables (where the mappings and connection states are kept), then indeed they can learn about your internal IP.
They can also learn about your internal IP if you actively advertise it to those you connect to. This can be happen in numerous ways. An application may simply read your internal IP from your network device and forward it along when connecting. And this is how probably you've seen this happen. With Java enabled, for instance, a browser can unwillingly and without your knowledge send your internal IP by running a malicious webpage script.
Note however that there are also legitimate uses for this, like recording internal and external IPs for purposes of abuse control and reporting.
However, the internal IP is of no practical use to anyone outside. They can't use it to access your machine. So this shouldn't worry you in any way.