Can someone confirm how to wire this air pump?
I have the 115V unit.

According to the data sheet, that is incorrect. There is no different model for 230V of this part number. You could rewire it for 230V operation just by disconnecting the wires and reconnecting them for 230V.

I understand that the brown and yellow wires go to the capacitor.

Correct.

Is the white wire considered my positive (+) wire where current flows in?
Is the joined blue and black wires considered my negative (-) wire where current flows out? The blue and black wires already come twisted together at their tips.

This is a non-polarized, AC device. There is no positive or negative. But these are the two lines for connection to your Hot and Neutral wires. I do not believe it matters which is which.

Do I simply ignore the orange and red wires? They too are already twisted together at this tips. Do I just put a cap on the twisted end?

If they are bare, then you should cap them for your and other's protection.
Additionally there is a ground connection screw hole you should connect to the ground wire.