CBSE had set an upper age limit for NEET as 25 years for Unreserved category and 30 years for Reserved. Although students above the age limit were allowed to attempt NEET 2018 because of the judgement by supreme court, you'll have to check for the age criteria of NEET 2019 again as the upper age limit might be capped.

AIIMS and JIPMER entrances have no upper age limit

You should have done your +2 (class 11 and 12) with Physics, Chemistry and Biology as compulsory subjects.

For complete eligibilty criteria please go through 

NEET Eligibility 

AIIMS Eligibilty

JIPMER Eligibility