There already is a bypass. It's called Route 481. Obviously it's longer, but it gets you past the city without ever going through it.
The long-term problem is salt, massive amounts of which get dumped on the highway during the winter to keep the highways passable when it snows.
That's what has caused the current overpass system to rot out. They do lots of maintenance on it every year. Right now, there's a major construction project on 690, the elevated highway that crosses I-81 in Syracuse.
A tunnel system would have a salt problem too. They'd have to use it even in the winter, at least at the entrances.
Perhaps re-building the whole elevated system with stainless steel might work.
/s