Similar to the Sky Force Anniversary game, the tournament is back on weekends. Beginning on Fridays (you can check the countdown clock for exact time), you are given a chance to take part in an endless level and survive and score as much as possible.
No real progress is required though. As soon as you start the tournament, the achievement will unlock.
Also similar to SFA, you need to complete Levels 1 through 3 to unlock the tournament mode. It shouldn't take real long to get there so even if you start on the last day of the tournament, you should be able to get there to complete it. Worst case, you can do it the next weekend.