Places that you can club for this trip would be Shravanabelagola. Melukote to Shravanabelagola is about 45 minutes drive / 35kms.
Places to Visit in Melukote
There are some historical places to visit at Melukote.
Early in the 12th century, the great Srivaishnava saint Ramanuja took up his residence and lived in this location for about 14 years. It thus became a prominent centre of the Srivaishnava sect of Brahmins, who obtained from the Hoysala king Vishnuvardhana, who had become a follower of the Acharya, an assignment of the fertile tracts of land in the neighbourhood, especially of the Ashta Gramas, on either bank of the Cauvery.Source
There are no specific timings if you are planning to just see the places. The best would be reach during the early hours or evening to avoid the heat of sun. Moreover the sunset view should be great which we missed.
The temple Pooja timings would be there, but other than that there are no restrictions. Plan your trip in a way that you should leave by sunset.
Cheluvanaryana Swamy Temple
The way to this temple is bit narrow if you go straight, take the right turn after the KSRTC bus stand, the road is wider. We did the mistake of going straight, the road were really narrow.
Yoga-Narasimha Swamy Temple
Another temple on top a very big rock. You will have be charged up with full energy. There are about 300+ steps if you are walking from the main road. Other option is to go up taking a left turn just after the KSRTC stand. The roads are narrow, but we could drive till the top without any issues. From this location you have climb about 120 steps.
View is excellent from this place. Beware of monkeys, they would sometimes snatch your bags or eatables if you have.
You should not miss this place, plan to be here during the sunset, you would really enjoy.