BY FLIGHT : There is no direct flight come to Srinagar by Mysore In this case you take multiple flight i.e. from Mysore to New Delhi & New Delhi to Srinagar.
BY TRAIN : There is also option to come by train. There is no direct train from Mysore to Jammu so in this case first you take train from Mysore to New Delhi i.e. Swarna Express & than New Delhi to Jammu Tawi it take 45 hours to reach Delhi and 10 hours more to reach Jammu. The major trains are Jammu Rajdhani express , Uttarsampark Kranti , Jhelum Express , Duranto express , Shiv Shakti Express , Shalimar express, Jammu Mail , Andaman express , All Jat express , Malwa express , Durg Jat expressthe total distance is 3250 kms but train drops you at Jammu Tawi Railway and than you choose taxi for further journey.
BY ROAD : There is no bus service or operator from Mysore to Jammu Kashmir. You take bus from Mysore to Delhi ISBT i.e. 2300 kms 34 hours to reach delhi & than take another Bus from ISBT Kashmiri Gate Delhi to Jammu Bus stand the approx distance is 600 kms & it will take 11 hours approx to reach Jammu by road. In Jammu there are various option to reach Kashmir you take local sharing Cabs local bus service and individual cabs . Bus service operates in the night only you take Volvo buses as well as local buses from Jammu Bus stand & rest services like individual taxi & sharing cabs you take from exit point of Railway station , Jammu Bus stand & Gumat chowk they operates whole day. The distance between Jammu to Kashmir is approx 300 Kms & it will take 8 to 10 hours to reach Kashmir.