I've got 2090 as category rank in Neet mds 2019... Can I register for counselling now... When is the last date

Yes, you can. The counseling for the first phase starts on the second week of June, and registration normally closes around a week before that. Hence, do it quickly.

The registration takes a little time too.

State level counseling will be conducted separately by state level authority, and you have to register separately for that.

Good luck!