We don't know very much about how the city of Byzantium looked like before Constantine refounded it and renamed it after himself. It was certainly much smaller than Constantinople. Its city limits probably extended from the Bosphorus up to the place where Constantine would found his great forum of which the so called burned colum still remains. The new city would have been three or four times this size.

Some of the older structures might have been destroyed like when Constantine build his great palace near the sea. But many others certainly remained, for example the old acropolis, the hippodrom and maybe the church of H. Eirene.

So yes, the old city was probably mostly left intact but also enlarged on a massive scale.