Hassan is about 110km (68miles) northwest of Mysore. You can easily get to Hassan from Mysore by road or rail. There are about 6 express trains from Mysore Junction (MYS) that travels via Hassan (HAS).
Train takes about 2 hours and bus about 3 hours. If you are driving by car you would cover it in just over 2 hours time.
Shimoga Express (Train No:16206) is a good option if you want to leave early in the morning. Dharwad Express ( Train No:17301) & Kannur Express ( Train No:16517) are late night options. The Mumbai bound Sharavati Express (Train No: 11036) is a weekly express train that pass via Hassan. Swarna Jayanthi (Train No.12781) is another weekly train via Hassan. There are also some slower daily passenger trains between Mysore and Hassan.
Regarding bus, the state owned KSRTC operates many (classes) of trips to or via Hassan from Mysore. You should be able find one bus connection in every hour from Mysore KSRTC bus stand.
Being a short trip, you would not require an advance reservation. However it is possible to reserve tickets , both in train and bus.
For schedules, reservation, fare etc please check the respective websites: www.irctc.co.in for train and www.kstrc.in for bus.
If you are driving by car to Hassan, the route is like this: Mysore city - 27 km --> Bilikere (deviation for SH57) - 13km ->  KR Nagara - 46km -> Hole Narasipura  -33km -> Hassan

Take the Hunsur road (SH17) out of the city. Many kilometers after Yelwal, you will reach the  Bilikere deviation for SH57. There is a huge signboard across the highway (SH17) directing to take the deviation for Hassan. Don't miss this modest looking junction, otherwise you are on your way to Hunsur instead of Hassan!  The road is in good condition all the way from Mysore to Hassan. Through traffic is relatively high on SH17 till the  Bilikere deviation, SH57 is pretty free of traffic.
You may also be interested in Mysore to Belur details.

on the way to Hassan

On the way to Hassan

on the way to Hassan

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