leveling up

depends, if you want to go hard on buying boosts and playing like 18 hours a day... like a week, otherwise playing a decent amount a day, could take a couple months

when you hit lvl 30 you unlock ranked. And at that point you also have acess to everything else you might not have yet (like runes/summoners, etc). From that point onwards leveling is not really significant anymore. The Level barely says anything about how skilled a player is. There are level 300 players in bronze and level 45 players in gold.

Why do you ask about lvl 100?

Depends on how much you play. Say you play 2-6 matches every day. You have a 50% win rate (for instance.) will take about a year.

https://leagueoflegends.fandom.com/wiki/Experience_(summoner)

^ if your someone who likes doing math i believe the algorithm is here.