To do this one quickly, simply play local 2-player arena matches 10 times in each mode: Free for all, capture the flag and tag.
Be sure to select RETRY after each game has finished to save a lot of time relaunching the arena game.
It should take around 45 minutes to unlock using this method, to speed it up, make sure controller one wins each game:
Free for all: position yourself on a crate and fire at the idle player; this is also an easy way to unlock Slime Attack - will take about a minute per match
Completing this first match will also net you Warrior
Capture the flag: drive to the flag and sit with it for 15 seconds 3 times to win. Should take a little over a minute per match.
Tag: park player one behind player 2 when the second controller is dropping coins and sit still when you are dropping them (there'll be a glowing shield looking thing when you are 'it'. This'll take about 2 minutes per game.
*EDIT* Use this website or a tally counter app to easily keep track of how many matches you've done
https://tallycounterstore.com/online-counter