Combine evaporative cooling with ventilation to increase efficiency of temperature equalization?
The way evaporative cooling works is that it forces air through moistened pads, evaporating the water and cooling and humidifying the air. The cooled and humidified air is then blown through the house and back outside again, lowering the interior air temperature and producing a cooling breeze as well. If I understand you correctly, you are asking if you could use evaporative cooling to use the heat already inside the house to evaporatively cool the rest of it. I don't think this will be any more effective than simply using outside air. After all, if it works, the temperature of the interior air will rapidly fall below the outside air temperature, eliminating the high temperature inside.

You have correctly identified that ventilation is necessary for evaporative cooling to work. You basically have two options: crack some windows, or use a dedicated ventilation system. Here in the southwest, there is a product called UpDux that is basically a vent that allows air exchange from the room to the attic above. They are used in warm-season houses that are evaporatively-cooled, allowing the cooled and humidified air to escape through the attic, with two benefits: 1) they allow you to keep your windows closed, and 2) the air cools the attic as well on its way out of the house, preventing the attic heat from heating up the interior so much. In your apartment, installing such products is not likely an option. I'd crack some windows in the rooms farthest from where you locate the evaporative cooler.

However, you have a potential problem: San Francisco is fairly humid for evaporative cooling. Less than the national average, but I see at http://www.city-data.com/city/San-Francisco-California.html that the afternoon relative humidity in July only falls to 30% in the afternoon, and it's higher than that most of the time.

You can figure out exactly how effective evaporative cooling will be using this calculator: http://www.srh.noaa.gov/epz/?n=wxcalc_rh

For example, plug in 90 degrees for the temperature, 30% for the relative humidity, and 30 inches of mercury for the atmospheric pressure. That will give you 67.55 degrees for the wet-bulb temperature. That's the theoretical cooling power of a 100% efficient evaporative cooler. Unfortunately, 100% efficiency is impossible unless you get a two-stage unit that is very expensive; assume more like 70-90%. Here is the formula for the real-life cooling power of a less-than-100%-efficient evaporative cooler:

Air temperature - ((air temperature - calculated wet bulb temperature) * efficiency)

So, 90 - ((90-67.5)*0.8) = 71.6. So an 80% efficient evaporative cooler, such as the Bonaire Durango window unit I own and love, would blow 71 degree air at 100% humidity into your house.

I can tell you from experience that once your evap cooler is blowing air that's in the 70s, it stops working that well, because the cooling power you're receiving is offset by the mugginess of the interior relative humidity rising to 60% or higher, which is also dangerous for a wood-framed house because that's approaching the mold-friendly level.

This may work anyway, since 71 degree air is certainly more comfortable than 85 degree air, but you'll need to keep it running constantly because 71 degree air is probably not going to cool the house down to below 77 or so, based on my knowledge of the heat gain you're receiving in a likely awfully-built, basically uninsulated San Francisco building with terrible clear-glazed single-pane windows. Additionally, you'll want to keep the windows open wide to ensure maximum air exchange to the exterior to keep the interior relative humidity down. If this is a plan worth pursuing, I would buy a hygrometer so you can measure the interior relative humidity for yourself.