Marshrutkas to Sevan town depart roughly hourly from near Yeritasardakan metro station in Yerevan. The journey takes about an hour along a good road through the mountains. Buy your ticket on the vehicle.
From Sevan town it is a 30-minute walk to the monastery, or a short taxi ride — around 1,000 AMD, though drivers will try for more. Agree the price before you get in.
Tour companies in Yerevan offer day trips combining Sevanavank with other sites along the lake or further afield — Dilijan to the north, or the khachkar field at Noratus to the south of Sevan town. These make sense if you want to cover more ground without dealing with marshrutka timing. A private taxi from Yerevan for the day costs considerably more but gives full flexibility.