My question is after steady state(ie. when capacitor is fully charged and behaves as an open circuit) is achieved what would be the potential difference across the dependent current source? How do I find it?
Assuming your v1 is constant and the current source 2v1 (times some suitable dimensions) is also constant your capacitor voltage will charge forever and the voltage across the current source will be approximately equal to the negative of that. It will climb until the current source headroom is reached or the capacitor (or other circuit component or connection) suffers a high voltage breakdown.