The title is the question. But I know Shankara mathas are the oldest, I'm not considering them for this question.
The term Advaita refers to its idea that the true self, Atman, is the same as the highest metaphysical Reality (Brahman).
Earliest reference to Advaita can be traced to Rig Veda Hymn X.72 and in Satapada Brahmana 6:1:1:5.
While in Rig Veda it was mentioned as Aditi and Daksha and in Satapada Brahmana it is was mentioned as Prajapati and AGNI.
Sri Gaudapada Mutt, also known as Kavaḷē maṭha (कवळे मठ), located in Kavale, Ponda, Goa, is the oldest matha of the Smarthan Saraswat Brahmins.
However, there is a controversy as to the birth-dates of Sri Sankara. The earliest period attributed was around 500 BCE.
All the Mutts established by Sri Sankara during his life time. If we go by this theory, then the Mutts Sri Sankara should be the oldest Advaita Mutts.