You can travel from the Ernakulam area to Kochi part of the city by city and by ferry.
Ferry is quick (20minutes) and cheaper (Rs6).
Ferry to Fort Cochin, Mattancherry , Willington, Vypin etc operate from here.
Ferries operate from the Ernakulam Jetty, near Marine Drive. It's 20 min ferry across the backwaters to the Kochi part. There are a few ferry stops, Fort Kochi , Willingdon Island and Mattancherry. You can also get ferry service to Vypin Island as well from Fort Cochin.
From Ernakulam Junction Railway (South station) walk to the Jose Junction bus stop (~1km walk). Take a direct bus to Fort Kochi (1 hour ). Or take a rickaw to the Ernakulam boat jetty (2km) and then take ferry to Fort Kochi.
From Ernakulam Town Railway Station (North Station), take a direct city bus to Fort Kochi (frequency 10-15 minutes). From the railway station walk 350 meters & cross the Banerji Road. Bus stop is near the Town Hall.
Or take city bus to Ernakulam Boat Jetty and take the ferry to Fort Kochi.
From From Ernakulam KSRTC bus stand, take a bus to Ernakulam Boat Jetty (2km) and then boat to Fort Kochi.
City Bus in Fort Kochi