It is a bit tricky, but this is the fastest way from Innsbruck to Rottach-Egern using public transport:
Train REX 5311 Innsbruck Hbf - Jenbach (train station) 10:21 - 10:47
Bus 8332 Jenbach (train station) - Maurach Mittelschule 10:54 - 11:18
Bus 9550 Maurach Mittelschule - Rottach-Egern Post 11:18 - 12:30
This is considerably faster than going by train via Munich, not to talk about that the train runs only to Tegernsee, from where you must go by bus or taxi to Rottach-Egern.
This is the only connection which will work (Monday to Friday), because there is only one bus on line # 9550 in the morning.
Although the two bus lines are operated by different companies, the possibility of changing in Maurach Mittelschule is noted in either written time table. Therefore I presume that bus # 9550 will wait for bus # 8332 in Maurach, as the layover is really short.