Take a look at Wyoming and Mississippi this past election - both had senators up for normal re-election (Cochran and Enzi) but both had lost a senator between Dec 2006-Nov 2008 (Thomas of Wyoming died and Lott of Mississippi retired). So Barrasso of Wyoming and Wicker of Mississippi, both appointed to fill-in for the empty seats, had to run for a special election in 2008. This allows them to serve out the remaining of part of the term which both happen to cycle in 2012
http://en.wikipedia.org/wiki/United_States_Senate_elections,_2012In April 1991 Harris Wofford was picked to replace the late Senator John Heinz of Pennsylvania. Now special elections don't have to be held on even number years but whenever a state has a statewide election, which the next one was in November 1991. Wofford defeated former governor Dick Thornburgh to keep that seat. But three years later he had to run again in 1994, which is when the seat normally cycles and he lost out to Rick Santorum.
So yes, Caroline Kennedy is going to have to raise enough money to run potentially 2-4 statewide races in the next 4 years. Not sure if New York has statewide races in 2009 - if they do her election to keep the seat might be held then.