I'll agree with skipping Yosemite this trip, because when you do go there you need to hike Half Dome. I've done it twice, it's amazing. 17 miles round trip past several waterfalls. Toward the peak it gets a little scary, but you'll be fine. It's beautiful, I highly, highly recommend it.
You can't go this trip because you need a permit and you'd need the cables to be up, and they are taken down right around now. When you do go back the secret is to camp near the trailhead and be on it at dusk. As you climb higher so will the sun, illuminating the valley below. You'll be on top by lunch time, beating any crowds. You'll be back down by 3 or 4 pm when the shadows are starting to get long. What a great day.
Since you won't be visiting Yosemite, your best bet is some hike involving Muir Woods/Mt. Tamalpais/Stinson Beach and some combination of those. It's 45 minutes from the Golden Gate Bridge to Mt. Tamalpais, not bad at all.
Big Sur is supposed to be beautiful too, and that's 3 hrs south of SF. You should do your research on whether that is worth a day to you (on the plus side I think you can take Pacific Coast Highway to get there).
And then inside the city my favorite place to sleep is the North Beach/Russian Hill/Nob Hill area. From there you can walk to North Beach, BART, the Ferry Building, Lombard Street, Coit Tower, Fort Mason, the Marina, and plenty of bars and restaurants. Bear in mind some of these may be 20 minute walks, and there are robust hills, but hey, you're on vacation. Or you can take a bus, no big deal.
I'd recommend visiting all of the places listed above, as well as taking a trip out to Haight-Ashbury.
As for runs, Presidio has trails, Golden Gate Park, Land's End, the Marina Headlands, and the Marina Promenade. My favorite run (it's long) starts in the Marina Promenade, goes under GG Bridge, down to Baker Beach, out to Land's End, through GG Park, and through the Presidio. But you can do portion of that obviously.
Some people like the Mission. I do not, though they have excellent Mexican food. The Castro is kind of a trip too, but you'd have to be with a gay friend and it sounds like you'll be alone. Fisherman's Wharf is not worth visiting except for In N Out burger.
If you do go to the East Bay, in addition to Redwood Regional Park is Joaquin Miller Park in Oakland, and Tilden Regional Park in Berkeley. These are all great trails for running & hiking, but it's probably not worth the trip until you've exhausted all the SF options, which you won't in 4 days.