How to determine polarity on cloth covered wires?
The wires themselves don't have polarity, that only exists within an outlet based on it being wired correctly, or reversed (hot/neutral reversed).

So essentially you just need to identify the hot wires and the neutral wires, and if you maintain this correctly throughout all of the wiring and correctly wire up your outlets, then polarity will be correct.

Non-contact testers are good for determining if there is any power on, but not perfect for finding the hot wire - sometimes neutral wires will give false positives if there is a hot wire in close proximity.

The best tool to use is a multi-meter.  The neutral and ground should read 0 (or a very low number), and hot to neutral or hot to ground should give you 120V (assuming your are in North America).

A standard outlet tester will tell you whether the polarity is correct or reversed. If it's reversed, then you just swap the hot/neutral wires.