I've heard this from my Japanese friend but I can't find any announcement or official ad regarding this discount.
Can anyone confirm if this is true or not?
While the English version of the USJ website doesn't note the special birthday pass, the Japanese language version of the site does:
You don't have to read Japanese to check out the page and see that you get a small discount. The one day-pass goes from JPY7200 (with tax) to Y6700 (with tax), for children from Y4980 to 4580 with tax. There's a similar discount for the 2-day pass (to Y10,288 and Y7241 for children with tax). You also get a special sticker and a character (Marilyn Monroe) will pose with you for a birthday photo.
There is also some fine print: you must be a Club Universal member (membership itself is free but needs to be done online). You can enter the park either in your birthday month or the following month. You can buy 1-day or 2-day passes for yourself and for 5 accompanying family members. You can order the tickets up to three months before your birthday month.
It looks like they go out of their way to make it difficult for non-Japanese language speakers to buy it, so I'd recommend asking a local friend to help. You can buy the tix online through their website or at the gate (but if buying the ticket at the gate, you will still have to pre-register as a Club Universal member and print out the coupon that they will issue you during your birthday month). I.e., you can't walk up without pre-registering and expect to get the discount.
While the official site doesn't mention it (that I could find) and as @JoErNanO said, some sites show a sticker will get characters to wish you a happy birthday, I did find this reference on tripadvisor:
small discount can be applied if someone in your party has a birthday in the month you visit, but you'll need to get a coupon in advance.
So it looks like you may be able to get a small discount, but it might not be worth the effort required. Your best bet may be to message USJ direct.
The universal studios Japan website carries no mention of discounted/free access on your birthday.
Searching on Google I found several references mentioning a birthday sticker which will cause everyone in the park to wish you happy birthday whenever they meet you. I couldn't find any mention of a free access on your birthday. However this blog mentions discounted access for the birthday-person and two guests:
You see a sticker on my skirt top? That sticker means its my birthday week and every staff in the park was wishing me happy birthday for that! Plus I had discounted tickets for myself and two people I brought along! Ain’t that great?