check out the ” Eastern Sierra Transit ” web site.
http://www.estransit.com
M-F there are bus trips from Lone Pine (McDonalds) to Bishop (Vons), leaving Lone Pine at 6:30am, 8:30am, 11:45am, 5:00pm. no Saturday bus.
also they run a Bishop Creek shuttle (Vons to South Lake) twice a day, leaving Vons at 8:00am and 4:00pm.
so if you can travel on Friday you could be up at South Lake on Friday evening, sleep in the bushes somewhere then start on Saturday. or there are a couple campgrounds not far from South Lake trailhead.
these 2 buses would cost you a combined $12 each.