Actually, the half-life of sulfur aerosols in the stratosphere is on the order of a few months. We would have to badly overestimate the amount required in order to put so much up there that waiting for it to fall out would be painful. And the proposal is not to put it up there all at once, but to do so gradually.
Nature puts a whole bunch of sulfur up there every couple of hundred years or so, through large volcanic eruptions, so we have a pretty good idea what the effects are. Besides global cooling, which of course would be the point, the major side effect is more colorful sunsets.
The most telling objection I've heard to the use of sulfur aerosols is that they do nothing for ocean acidification.