This probably isn't the right forum but was hoping someone could help. My partner has an Austrian mother but was born overseas. She would like to get her Austrian residency permit and live there.

How does one go about this?

Talk to the Austrian consulate in your country.

http://www.austria.org/citizenship

She needs a Aufenthaltsbewilligung (residence permit.

The first question is the reason why she wants to move to Austria. There are several possible reasons:

Rotational employee
Seconded employee
Self-employed person
Artists
Special cases of gainful employment
Pupils
Students
Community service employees
Researcher
Family community

For more details, see:
http://www.bmeia.gv.at/en/travel-sta...hite-red-card/

Second, there are general preconditions for the residence permit. The main point is that she needs a permanent monthly income of a least 872.31 Euros. For further requirements and documents, see here (German only):
https://www.help.gv.at/Portal.Node/h...te.120217.html

As for Austrian nationality, there are several possibilities. If your partner's parents were married to each other, she is Austrian as long as she was born on or after 1st September 1983. If she was born before that, the father had to be Austrian so she can't derive her citizenship from her mother. If the parents weren't married, she automatically inherits mother's nationality. She should check her exact situation with the Austrian embassy or consulate.