Does an O mint mark mean that the coin was struck before the P mint?

Not strictly, because that's really 2 different questions in one.

The New Orleans Mint used an O mint mark from 1838 to 1861 and 1879 to 1909, and the first coins to carry a P mint mark were "war nickels" minted from 1942 to 1945, so the O mint mark definitely precedes the P mint mark.

Because the Philadelphia Mint had been striking coins without using a P mint mark beginning in 1793, coins from that city can easily predate coins from New Orleans.