Your surname is Scottish in origin, not Irish. It is derived from an occupation (like Smith, Wright, Fletcher and so on, all of which indicate a particular trade or profession). Sout(t)er means a shoemaker or cobbler. The Latin word for that is Sutra, so it'll have come from a corruption of that.
In earlier times in Scotland people didn't really have surnames. The communities were small enough to recognise anyone either by their "forename" alone or by a system of patronymics (naming after your parents eg John MacDonald was John son of Donald.) Alternatively sometimes they were identified by occupations. (The Welsh still talk about people in that way. So "Jones the post" distinguishes Jones the mailman, from all the other Jones in the area.)
Many early Scottish documents would have been written in Latin, so if an early ancestor of yours had been a shoemaker he might have been Cliffordus Sutra (Clifford the Souter)in a legal document. Clifford the Souter would eventually became known as Clifford Souter when it became practice to acquire a surname, as populations expanded and officialdom needed to distinguish between people.
There's no tartan that I am aware of. Tartans are generally associated with the Scottish Highlands, and Souter is found mostly in other parts of Scotland ie the North East, the Lowlands and the Southern Uplands, where tartans were not really used.
Scotland's national poet, Robert Burns, wrote a very famous poem in the 1700s called Tam O'Shanter. In it he mentioned his real life friend Souter Johnie (ie John the shoemaker). Souter Johnie's cottage in Kirkoswald, Ayrshire has been preserved as a museum, and is open to the public. That's probably the most famous reference to the name Souter in Scotland.