That's almost exactly where I saw it last month. rear mezzanine center, Row E. The Richard Rodgers is a pretty small theatre, and the incline is steep, both of which makes rear mezzanine seem a lot closer than other places. The View From My Seat photo is about right.
Yeah, you're not going to get close-ups. But for Hamilton, you get the advantage of a panoramic view of the staging and choreography. There are several numbers (I don't spoil them) where rear mezzanine actually has its advantages. And if you're lucky enough to see Leslie Odom Jr., who plays Burr, he belts out his big songs to the folks in the rafters. (Don't you assume who you will see, though. People get sick (Lin-Manuel Miranda had laryngitis the day we saw it, although I did get to meet him outside the theatre as a result.) But as others have noted, the cast shouldn't start turning over in large numbers until mid-July.
As birthday presents go, that sounds like a pretty good one. Enjoy the show!