So he would run for governor in 1997, win re-election in 2001 because let's face it, Gore isn't going to tap him for VP nominee. Then he could run for the nomination in 2004, but I think Springsteen is going to want to go the Reagan route and finish out his full second term.
Now the problem is that in 2008 he knows that he'll be running against Hillary unless some massive butterflies intervene. His relationship with the Clintons is pretty good from what I can tell and it will only be better if they helped him run for governor in 1997. So if there isn't some way that Hillary isn't running or there's a giant schism between the Clintons and Springsteen I'm not sure he'd actually run.
Supposing that he did run in 2004, though, I don't see a whole lot of obstacles to him winning the nomination. The problem will be that he's not going to have a whole lot of gravitas in an election that will revolve entirely around foreign policy even if he picks someone like John Kerry or Wesley Clark as his running mate. For my money, I think he'd probably lose to Bush.
Now, maybe he runs for governor 8 years earlier in 1989, wins, serves until Bradley announces his retirement in 1996, and then runs for the senate in that year. If he wins that race and is re-elected in 2002, I think he's a much stronger candidate in 2004 and might have a shot at winning.
Nevertheless, Springsteen as president would be very disappointing. I like his music too much to want him to sell out and be a part of the system (even more than he already has).