Better halve that to 90 hits, cause putts are "free" and don't count as hits. Still sucks though...
Actually, a little more than that, probably.
echbot0's GIR is about 70%. So 30% of the time he'll have (at least) one more ball hit. Over 45 holes that would add about 13-14 ball hits on average. So, realistically, about 103-104 ball hits, or about 1 full ball to play a Showdown start-to-finish (depending, of course, on the ball's durability rating).