It's based on days of service time, which you accumulate by being on the big-league active roster or disabled list. Yes, September call-ups do accumulate service time.
But unless they were planning on holding him down the next year to try to slow down his arbitration status, the September service time wouldn't mean much. Once you send them back down again, they stop accumulating days of service time.
If you have three full years of service time (1 year = 180 days) at the end of a season, then you get to go to arbitration. If you have less than two years, you definitely cannot (and the team can just pay you whatever they want, usually the minimum or close to it).
If you have between two and three years, you may be eligible for arbitration or not. They take all the players between 2 and 3 years, and the top 22% of them get to go to arbitration a year early. These guys are called "Super Twos." Last year, it took 2 years and 134 days to get Super Two status, so about 1.5 months short of a full year.
So if you want Baez to avoid Super Two status, you have to hold him down for about two months in the season you bring him up for good. If you call him up the September before, then every day he's on the active roster is an extra day you'd have to hold him down the next year to avoid Super Two status.
Similarly, there's the concern of when he becomes a free agent. If you have six full years of service time at the end of a season, you can be a free agent. 5 years and 179 days, you have to wait another year. Holding a guy down for a couple of weeks in his first year is usually enough to keep him for 7 years instead of 6 before free agency. Same as before, every day you have him in September is a day you'll have to hold him down the next year, if you care about such a thing for that player.
The final concern is options. An option year is one in which the player is on the 40-man roster but not the active roster for any part of the year. Most players get three options years (there's a complicated reason why some get four), but that won't matter for Baez unless he's not as good as we hope he is (he's not going to spend three more years in the minors, hopefully, after being added to the 40-man). So calling him up in September won't burn an option, because you are doing it to put him on the active roster anyway.
Long story short: A September call-up (though very unlikely for baseball reasons. The kid hasn't even hit above A-ball yet in competitive games) probably wouldn't cause any service time issues for Baez.