We were in Istanbul in June, there was absolutely no problem walking around on your own. We took a free tour, which was great.
It´s not difficult to get in or out of the cruise port but the walk through the building itself is loooong and we found it annoying. They put up turnstiles and close glass doors so you can see people zig zagging through the building.
The tram stop is maybe 200metres /5min max on the main road after exciting the cruise port.
When using the tram you of course need to buy a ticket. Be aware that the machine only accepts credit card, smaller turkish banknotes or coins.