There is no direct connection from Batumi to Erzurum. However, you can take the bus to Hopa, take the bus to Artvin, take the taxi to Oltu, then take the bus to Erzurum. Alternatively, you can drive from Batumi to Erzurum in around 4 h 46 min.
From Batumi bus to Hopa, bus to Artvin, taxi to Oltu, then bus to Erzurum. 7 h 37 min 567 ₺. From Batumi drive to Erzurum. 4 h 46 min 193 ₺.