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, you can inspect the excavated residence of Iverian rulers, including bathhouses, a royal sarcophagus, a wine cellar, a six-apse temple and what may have been a palace.
It's 3km from Svetitskhoveli – back across the Mtkvari bridge, then 900m along the Tbilisi road to an 'Armaztsikhe-Bagineti' signpost, from which it's about 800m along a path to the site.