You can actually check your progress on the Fame front whenever you're recording for the job you got. If you look at the sequence, beside the amount of money you gain from the job, you'll see the amount of Fame you currently have. To be able to win the Seiyuu Award, you have to get at least 500 points.
Some tips that could help: (1) Take the job from the journalist in the bookstore when you meet her requirements; I think she can give you +20 Fame points despite the meager amount of money she pays you. (2) If you want to be able to spend more time with Haato's love interest, buy a computer and a shower/bathtub from the furniture store and use them at night so you don't have to constantly take extra VA classes, even just a little bit. Using those items during Sundays can net you more points than if you do them during nighttime. (3) Keep saving before auditions so you can get more jobs. Also, since the point system for this is random, you can even save before lessons if you want to change the amount of points you get.