Somnathpur (Somnathpura) is a village about 35km east of Mysore City. This is one of the most popular day's excursion site from Mysore.
The 700 years old Chennakesava Temple located on the banks of Kaveri is is boasted as the most complete speciman of the 'Hoysala' style architecture surviving today. This is in the same league of style of the temples at Belur and Halebid.
Belur Halebid circuit probably need a full day to explore (considering the distance from Mysore). In that context Somnathpur is pretty close. You can reach in about an hour's time from Mysore.
Opt for some sort of private transport, as the public transport, though available are a bit infrequent.
The temple is under the care of Archaeological Survey of India (ASI). Somnathpur temple is open to visitors 9 am to 5.30pm. The standard nominal entry fees is collected at the counter next to the main entrance.