The counselling process for NEET exam all over India in all the states is one and the same. You have to do the document verification and then submit your colleges choice filling list. For admission in Karnataka being a candidate from Maharashtra, you will need to apply under the 15% AIQ quota reservation. Around 564 marks was the AIQ quota cutoff last year in private colleges of karnataka equiavelent to an AIQ rank of 12763. Having said this cutoffs keep fluctuating from year to year as they depend on various factors like number of candidates appearing and also the paper level difficulty, etc. So try to score above 580/590 to be on the safer side.

Refer to the sites below for more detailed information..

https://medicine.careers360.com/articles/neet-cutoff-karnataka

Best wishes. Thank you.