You can arrange to see Alcatraz on your own. The Alcatraz tour boats leave from a pier not far from where your ship will be docked and a tour of Alcatraz shpuld be about 2-3 hours, and at much cheaper than $67.00 per person.
Alcatraz is a DO NOT MISS trip for several reasons including the boat ride to and from the island.
Toward saving some money you can see much of San Francisco by walking. From the pier you can follow Columbus Avenue to the North Beach area. There are several wonderful Italian restaurants here, the Peter and Paul church where Joe Dimaggio and Marilyn Monroe were married and the City lights bookstore. You can also walk from here through Chinatown and onto downtown San Francisco and take the cable car back to the pier (the Hyde Street line is more interesting but both lines will get you back to the pier area (called the Embarcadero). For lunch there is an Inn and Out burger near Fisherman's Wharf for a very good but inexpensive lunch. For some atmosphere have an Irish Coffee (Alcohol)or a plain cup of coffee at the Buena Vista Cafe which is on Northpoint and Hyde.
Remember, San Francisco is not very big and you can reach the Embarcadero easily within 30 minutes even in heavy traffic.
I would do the Alcatraz trip in the morning and spend the balance of the day walking around San Francisco. And if you have time the Golden Gate Recreation Area is within a nice walk from where the ship docks.
Hope that this helps.