Mtskheta was an important place as capital of the Iverian kingdom well before St Nino arrived. At Armaztsikhe-Bagineti, on the south side of the Mtkvari River, you can inspect the excavated residence of Iverian rulers, including bathhouses, a royal sarcophagus, a wine cellar complete with kvevri (Georgian wine amphorae), a six-apse temple and what may have been a palace. The earliest remains date from the 4th century BCE, though the site is still far from fully excavated.
It's 3km from Svetitskhoveli – back across the Mtkvari bridge, then 900m along the Tbilisi road to an 'Armaztsikhe-Bagineti' signpost, from where it's about 800m along a path to the site.