Illinois politics has long been the most insider friendly quid-pro-quo environment (with the possible exception of DC, at least) in the nation. This whole Blagovich thing is a surprise to no one, and kudos to the local dems for keeping him out of the press just long enough.
His dirty political stench will absolutely impact Obama's first days in office.
If Obama is a member of the Illinois democrat inner circle that got him into office, he should encourage a delay in filling the Senate seat he left vacant, so the new Governor can whisk another Democrat insider in. If Obama wants to distance himself from that cesspool and demonstrate he's setting aside partisan politics for the good of the nation, he could very publicly support a special election for the seat. Chances are a Democrat would win it anyway, so the actual political risk is minimal to him.
If he did call for a special election, I would be very impressed by the move. If my character assessment of him is accurate, though, he'll lay low and let politics as usual prevail---why run the risk of losing the senate majority? Please note---my character assessments aren't 100% accurate.