That sounds sweet as heck!
Well I mean, really I was too lazy to do the research Giles did. So, what he said! 
Assuming I had a babelfish, I would want to have a long talk with Pythagoras.
I want to go hang out with Herodotus. I bet he was a kick after a couple of drinks.
Lovely. adding Eleusis to itinerary
ETA: It’s weird, but although Pompeii is one of my Most Amazing Places in the modern world, I don’t have much inclination to go visit it when it was alive and kicking. I’m trying to figure out why.
I love Roman and Greek history but something tells me that I wouldn’t fit in well with the Greeks and that I wouldn’t be able to stomach the Romans.
Chinese history is in an Interesting period, but a turbulent one also (Warring states AND 3 kingdoms fits into the time frame as well as the fall of the Han) though I’m sure you could find a 20 day period there in the midst of the Han prosperity where you could have a hell of a time provided you brought enough gold.
Buddha died in 483 BC, so that misses the cut off, but good luck tracking him down even if it was in the right era. You’d be in the sweet spot to witness Ashoka’s reign though.
The obvious answer would be to peep in on Jesus, and ultimately I’m assuming that’s what everyone would choose once in the buggy but as Dopers we all want to be a smidge unique. I’d choose hitting up Britain post-Caesar. Gorgeous land, simple people, soothing baths, hot Celtic women… with a satchel of Gold and a few robo-guards making sure you aren’t daggered in the back, you could literally live like a king without worrying about the sanitation of living in Rome, or the super-hetero-man-friendliness of Greece, or the pressures of living in a non-AC world in the middle East, or living with people who literally did not know how to sew cloth in Japan…
I’ll take a babelfish, an invisibility generator and a lavish camp set up (the self making and breaking kind). I’m going to North America to look at the pre-columbian Native Americans. I’m just going to spy on them, I’m bad at meeting new people in the best of circumstances. I’ll probably skip around the continent focusing on the prettiest places.
She walks like a woman and talks like a man?
I’ll sacrifice the two weeks for a hyper-dimentional cabinet into which I and Hypatia and her colleagues can put a copy of everything in the Library to bring forward to the present day. Then I’ll bury it all in the sands of Namibia or Mauritania and ‘discover’ it, ‘led on’ by an ancient scroll in Hypatia’s own hand. I’d have chosen a cave by the Dead Sea, but they’ve all been taken, and it’s important that they be discovered after my return so as to not disturb my timeline.