If you follow the narrow laneways around the Old Town towards the old thermal baths and Tbilisi mosque, you will eventually arrive at the stairs up to Narikala Fort. It looks difficult but you can expect to get up the top within 10-15 minutes and the views are worth it. The fortress is the remains of a 4th-century bastion that is mostly crumbling and free to roam around in. There is a restored church inside the walls as well. If you continue along the ridge towards the west, you can follow a path that takes you to the Mother of Georgia Statue.
This is the best spot in the city to watch the sunset and capture the perfect golden light over the cityscape. In fact, I think it’s one of the best free things to do in Tbilisi.