Almost all the bus going to Udupi/Manipal from Bangalore go via Mulki. There are at least 100 buses from Bangalore to Mulki every day. I personally have good experience with Pragathi and Vishal buses.
You can take the train if you are coming a day early. Otherwise please take bus/flight. Mulki is also connected to Bangalore via Rail. Train Number 16523 reaches mulki at 10:30 AM.
You can take flight to Mangalore to reach Mulki. Two early flight leaves around 7:00 AM from Bangalore. Mangalore is around 40 minutes by air.