Should i do the game again for the waifus and the bros ?I'm trying to start a romance with Futaba but i'm already at december so i don't know if it's really useful

Hi everyone, so i'm actually near the end of Shido Palace (i think) and i wanted to complete my first game of Persona by doing the true ending. I don't remember every condition but i do remember that one of them was related to Mémentos. Do i need to reach the deepest part of Mémentos before the end of the Palace ? And what are the other conditions ? (sorry for my bad english)

I finally got the opportunity to buy and try Persona 5 , and i really liked that game but when i say Persona 5 Royal i got really curious ... i saw a few trailers and i have to admit i'm a bit confused ... is it like .. a Persona 5 "Remake" ? (sorry for my bad english)