Among many other things, October 1 is Nigeria's Independence Day, marking Nigeria's independence from British colonial rule in 1960. This is a day of celebration, so "living every day like it's the 1st of October" suggests the speaker likes to party, or to treat each day as a celebration of life.
Another line from this remix of the song supports this:
I'm fresh from Lagos, fresh from Nigeria
Olushola is also a common male name in much of western Africa.
"Party like it's (time)" appears to have originated in Prince's 1982 song 1999 which contains the lyrics "party like it's 1999," referencing the massive turn-of-the-millennium celebrations anticipated at the time.