I saw on the Winter Bash site that I could earn a hat for posting a question (+1) on meta, and also that I could earn a hat if I answered my own question. But my question is now meta: Can I get the hat for answering my own Meta question?
The hat you are referring is Constantinopolitan, which is given in the main site for a question you posted on the meta site. That is the only hat you gain for posts on the meta site. The other hats are for posts on the main site.
ask a question or provide an answer on meta with a score of 1 or more
The other hat is Git R Done.
Git R Done
ask and answer your own question, each with a score of 1 or more
The only exception about meta sites is Meta Stack Overflow, which is a site of its own, the meta site for Stack Overflow, and it doesn't have its own meta site.
On Meta Stack Overflow, you can get all the hats but Costantinopolitan.