Can a nickname for Anastasia be Nate??

I don't think so. I mean, when you hear Nate you don't think, oh, your real name must be Anastasia. It sounds more like a shorter version of Nathan/Nathaniel.

You can nickname someone whatever you want. It doesn't necessarily have to relate to their given name, although that's a very common way of nicknaming people.

Sure, it could be if you want it to be. Lots of nicknames make you go, "How did they get THIS from THAT name?!?" (Take Peggy as a nickname for Margaret as just one example!) Nate for Anastasia is a bit odd, but not as strange as some already-established nicknames.