The guidelines for departure from Pune can be found here: https://indiaairport.com/airports/pune/passenger_guide.htm
Note that you will need
• Tickets with corresponding dates
• Identity card / Passport
It does not mention anything about the entry time into the airport and recommends to be at the check-in counter 2/3 hours before departure.
Personally I found airport entry controls to be very inconsistent and hard to predict. During Covid there were some airports that did indeed have a limit on entry time, but I think this has been mostly relaxed. Sometimes security will hold passengers outside if they feel it's too full inside the terminal.
I recently flew from both Mumbai and Delhi (but not Pune): Mumbai was ok, Delhi was a disaster. At least half of the people in front of us had "issues" and were held up considerably or turned around altogether. At some point they just closed the door completely with no explanation at all. My partner had a spirited conversation with the remaining security guard and he reluctantly opened it again.
I'm guessing that most of the problems were related to what exactly constitutes a "ticket" specifically with online bookings, confirmation emails, offline PDFS, etc. We had a boarding pass and passport, which worked fine. So I would recommend checking in for you original flight online and bringing both an electronic and a paper copy of the boarding pass.