Welcome to LOC.
A new service book is no use to you without the service records. Try your local Lexus dealer and see if they can get some information for you either on when it has been services or who by.
If you have no SH as a minimum you need a service with a cam belt change, if the belt goes so will your motor.
I'd get it running right before you spend any money on it making it going faster.
Personally, a low mileage car with no SH smells a bit :( Hope I am wrong. Where did you get it from ( eBay, private, dealer ) ?