The option to naturalise and take Austrian citizenship only exists after a minimum of 6 years of continuous residence in Austria, and for Britons who have not been in Austria that long, there are two other pieces of paperwork what should be done to "regularise" your right to remain indefinitely. Having done the Anmeldung upon arrival, after three months, you need to do the Anmeldebescheinigung, which effectively is tied up to the EU's Freedom of Movement. In Austria the Niederlassungs- und Aufentshaltsgesetz transposes European Directive 2004/38/EC. Once you have done the Anmeldebescheinigung, and doing it on time is important to avoid a fine. After five years lawful residence you may request a Bescheinigung des Daueraufenthalts - not yet mandatory but good for peace of mind - you submit your application through MA35 and it is issued by post usually within a few weeks, but will not automatically pop through your letterbox.
Following my querying the claim that Britons would not need to apply for permanent residence on the Ambassador's recent Facebook Live Chat, a clarification was given by the Embassy.