For a driving excursion en route to Ani, you could head to Bagnair Armenian Monastery ruins. Two church building-shells remain, the larger one (10th century) is badly damaged; the smaller one, 200m across a gully, is in better condition, with a nice 12-sided drum-tower adorned with blind arcades.
They lie at the western edge of tiny Kozluca, a Kurdish village 14km west of Ani. From the Ani–Kars road, turn south at the western end of Esenkent, then, after 6km, turn west and drive 2km into Kozluca.