Assuming that your guy has less than 15 days previous MLB experience, you'll have to wait for 20 days from the beginning of the AAA season.
Please note that he has to be on an MiLB roster for at least 20 days in order to prevent the entire season from counting towards ML service time. The AAA season is what counts - so since the MLB season typically starts a day or two earlier, you have to be careful which schedule you look at it.
Also, remember that it's HBD days that matters; not games.