Should I connect a new ceiling fan as the old one was?
I think the switch was originally on the neutral (downstream) side, which should be corrected. Here's your correct configuration:

Cable 1 is the supply (home run). Cable 2 is the switch loop. 


Blue and black from the fan should connect to 2-black (the return from the switch).
White from the fan should connect to 1-white (the source neutral).
The remaining black (source hot) should connect to the remaining white (switch supply). 


It's pretty much this, but you'll have a blue along with the black from the fixture.