From the Fourth of July TH, hike up the trail that eventually leads to Arapaho Pass. There is one switchback right at the beginning and then another more pronounced switchback about 1/2 mile up. The trail is well-used, easy to follow and ascends at a fairly gentle gradient. At 1.6 miles up, the trail comes to the Fourth of July mining ruins. (See coordinates provided.) The trail for the Arapaho Peaks turns of in this vicinity and was signed when we passed by in 2012. You may see other trails earlier that serve as a shortcut. Most likely, the Forest Service discourages their use.
From the trail intersection, head up past low trees and onto open tundra slopes. The trail heads NE and E as if climbs steadily on switchbacks through mostly tundra terrain that gives way to more rock higher up. Near the saddle between South Arapaho and "Old Baldy," (UN13,038) the trail joins the "Glacier Trail" coming in from the east. At the saddle, turn and head NW, basically following along the ridge to the summit. The gradient increases in steepness. There are spectacular views looking down to the right into the Arapaho Glacier cirque. Staying on the trail will keep you safe. Venture near the edge at your own risk. The trail finishes amid boulders & rocks at the summit.
Now it's time to take a good look at the weather and decide if you have the additional two hours minimum to make the traverse on over to North Arapaho and back without putting yourself at too much risk. To descend from S. Arapaho, simply retrace your ascent route back to the TH, but don't forget to include "Old Baldy," which is no more than a 15 minutes jaunt up from the east saddle over mixed tundra & rock.