Service time is calculated based on the regular season, so the postseason will have no impact on that. To manipulate service time next season, you will just need to wait another day (so wait 21 games instead of 20 if you bring him up after Game 161).
As for adding him for the playoffs, is he already on your 40-man roster? If not, you might run into an issue there. In real-life MLB, I know you can do add a player who isn't on your 40-man for the playoffs, but not sure if it works the same in HBD. One thing to do is to make sure to add him to your playoff roster since he wasn't up when rosters expanded. Don't just think he is on there because you promoted him to the majors. The playoff roster add is an additional step at this point in the season. Side note: You might want to put him on the 40-man while he is in the minors and see if you can add him to the playoff roster before promoting him. That won't affect anything for next season.