You might be able to rent a bike from these guys. http://www.the-bike.ch/
They are in Pully (east side of Lausanne). You can see the test bikes (they may rent them?). You will need to tell them your height. You can write the email in English as the owner is USA and the chief mechanic is UK.
You should also have a helmet (bring with you?). People here do not mountain bike without one.
For the rest, you will hopefully get some suggestions from others, the search function or one of the tourist sites.