The Monastery of St. Barnabas is at the opposite side of the Salamis-Famagusta road, by the Royal Tombs. You can easily tell it by its two fairly large domes. It was built to commemorate the foremost saint of Cyprus, whose life was so intertwined with the spread of the Christian message in the years immediately following the death of Christ. http://www.cypnet.co.uk/ncyprus/city/famagusta/stbarnabas/index.html

This was originally a church built in 1359 with funds donated by a successful merchant called Simon Nostrano during the reign of Pierre I. http://www.cypnet.co.uk/ncyprus/city/famagusta/mq-sinanpasha.htm