I'd go for Sweeney rather than Phantom. I've always found Phantom to be a mediocre show and a tourist trap, while Sweeney may interest you because of the actors playing the score onstage. It's much more challenging material as well.

You might want to consider the Empire State Building since they just extended their hours until 2am.

Sweeney Todd is a great choice and will be very different from Spamalot. I've seen both and loved them both.

Also, as budgetqueen posted, Empire State Building is now open until 2 am and recent news reports suggest that it gets a number of visitors after midnight. There are pictures on the NYTimes website and it looks like a spectacular nighttime view.