As tourists, British citizens can stay in the Schengen AreaSchengen Area The Schengen Area is European border control-free travel area, consisting of all EU Member States except for Bulgaria, Croatia, Cyprus, Ireland and Romania, as well as Iceland, Norway, Switzerland and Liechtenstein. (including Austria) for a total of up to 90 days in any 180-day period, visa-free. Days spent in any Member State in Schengen count towards these 90 days (e.g. a holiday in Spain).
To be able to stay longer than 90 days, or to work in Austria, you will require a visa or a residency permit. Dual nationals (i.e. holding a British citizenship and the citizenship of an EU or EEAEEA The European Economic Area (EEA) consists of the 27 EU Member States as well as Iceland, Liechtenstein and Norway. The United Kingdom is not part of the EEA. country) can stay for longer by entering on their EU/EEA citizenship.
Marriage to an Austrian or EU/EEA national does not confer you any special status and you will still require a visa/residency permit.
Overstaying the 90 day limit in a rolling 180 day period could see you categorised as an “overstayer”, which can be punishable by a fine and a ban from entering the entire Schengen Area.