On my trip to India I spent 2 months touring the entire country and I only came to Hassan so as to access the temples at Belur and Halebidu.
Hassan is a practical stopover if you want to see these temples, but otherwise it's really ugly! It's just one big intersection that has cheap places to stay. So if you are planning to stay in the region for a while, then I suggest you stay at Belur or Halebidu. Further along from these sites is Shravanabelagola, another nice and calm place.
In brief, like other travelers who arrive here, I followed the standard pattern - I arrived, I toured for a day, I slept over and I left. There is simply nothing to do in Hassan itself.
As you will notice during your stay in India, there are cities whose only assets are their geographical locations. Hassan is one of them. Its advantage: being the epicentre of the stunning cultural triptych of Halebidu, Belur, Shravanabelgola. And I'll talk about it in a moment.
Only 50 km from Hassan, in a Jain temple built on a hill at Shravanabelgola, stands a gigantic statue in honour of Gomateshwara, a saint also known as Bahubali. From its height of 17 metres and its thousand years of existence, the stony faced giant surveys Karnataka in clothing reduced to a minimum, provided that the liana which surrounds its limbs is considered clothing. The site is magical and the view is fantastic! (good luck with the 600 steps!)