Dear Rahul, if your question is whether you can apply for NEET 2019, yes you can. The eligibility criteria is that you should have been minimum 17 years of age atleast by December 31st 2019 and you should have studied Physics, Chemistry, Biology/Biotechnology and English as core subjects with 50% as qualifying marks if you fall under general category. If you fulfill this criteria, you can apply for NEET.

For AIIMS you should have completed 17 years of age on or before December 31, 2019. And you should have scored 60% in your +2. But it doesn't talk anything about age gap. So I would suggest you to register for Basic Registration with AIIMS and you will know whether you will be eligible or not.