There is no limit of Attempts, Only age limit is there 25 years for General Category & 30 years as a reserved category. You can apply for NEET Entrance exam if you meet the eligibility. You can check the completele eligibility with the given below link.

https://medicine.careers360.com/articles/neet-eligibility-criteria

Good Luck!

Ofcourse you can try for neet 2019 there are 3 attempts in neet , if you are trying 2nd or 3rd time then their is no problem to take 2019 neet

NEET 2019 can surely be given as maximum 3 attempts are allowed. Which means if your 2018 was first attempt then you can give NEET 2018, 2019 ajd 2020.

List of all eligible parameters below. 

https://medicine.careers360.com/articles/neet-number-of-attempts-and-age-limit