I drafted 89 Milacki (Progressive)  and kept him stashed for most of the year...bringing him up to close at the roster deadline. He has not reached the 95% mark for IP. Play-offs start...I am the WC...good team with 99 wins.

Gooden-Ryan-Walk-Welch regular rotation

Should I keep him closing? Bullpen is solid, so there are others I could easily slide into close.
Should I start him game one of the 5 game series and try and steal a game vs. the #1 seed?
Save him for a must win game?

How many starts could I expect from him in the play-offs?

I'd use him in the first series only if it were a "do or die" game, and then hope Gooden is rested to start the final game, or if you find yourself blowing series lead and end up in a game 5, start him there. If you go deep in the playoffs, he should be good for a second game at over 90%

Ideally you use him as a closer and then start him  game 7 of the world series. unless an earlier series goes to the final game then start your best. If you'd like to start him twice consider giving him a low pitch count for the first start. and don't assume you'll be "stealing" a game there's no guarantee he'll win.