To unlock the Model Warrior Julianne minigame, you must purchase both the "PC-9X" and "Assorted Cartridges" from the shop. Once you have both of those purchased, click on Jill's computer in her room to start the minigame.
There is a pretty easy method for this minigame, which involves hiding in the bottom right corner of your screen for a good chunk of the waves. Refer to the video below that I've made to show the best times to dodge/kill enemies, and a very easy method to kill the boss.
Timestamps:
0:52 | dodge enemy
0:58 | dodge enemy
3:26 | dodge enemy
3:33 - 3:40 | dodge 3 enemies
3:51 | shoot enemy
3:58 - 4:03 | shoot 3 enemies
4:27 - 4:32 | dodge 1 enemy and shoot 1 enemy
4:54 - 5:00 | dodge 2 enemies
5:50 | float above the boss and shoot
I have included a timer in the top corner of the video to help with the timing for your own run. Once you get to the boss, simply float directly above her and shoot for a very easy win.
Credit:
1. Thank you to 'Black Blood' in the comments below for providing the idea of adding timestamps on when to dodge/kill. I've changed them slightly to reflect the timer in my own video, so hopefully it's slightly more accurate that way.
2. Thank you to 'SGTGUNYOUTOHELL' for posting how to easily cheat the boss. Originally, my method was to stay behind her and pop out every once in a while to get some hits in. Floating above her like in the video is definitely the easiest, though.
I've combined aspects of the best advice/solutions and put them into a single video, I hope this helps!