This is easy to simulate. Starting with today's leaderboard, assuming no other games happen in the meantime and all games take place in the same day, it takes 31 games for a new player to get to #1, and 35 games to get to level 51 (by always winning a 2-player game against the top-ranked player that is not himself).
game 1: start level 0 (ranked #18056); win against Obi Wan Bonogi
game 2: start level 3 (ranked #10348); win against Obi Wan Bonogi
game 3: start level 6 (ranked #7268); win against Obi Wan Bonogi
game 4: start level 9 (ranked #5400); win against Obi Wan Bonogi
game 5: start level 12 (ranked #4125); win against Obi Wan Bonogi
game 6: start level 14 (ranked #3168); win against Obi Wan Bonogi
game 7: start level 17 (ranked #2437); win against Obi Wan Bonogi
game 8: start level 19 (ranked #1880); win against Obi Wan Bonogi
game 9: start level 21 (ranked #1434); win against Obi Wan Bonogi
game 10: start level 23 (ranked #1111); win against Obi Wan Bonogi
game 11: start level 24 (ranked #855); win against Obi Wan Bonogi
game 12: start level 26 (ranked #664); win against Obi Wan Bonogi
game 13: start level 28 (ranked #514); win against Obi Wan Bonogi
game 14: start level 29 (ranked #380); win against Obi Wan Bonogi
game 15: start level 31 (ranked #292); win against Obi Wan Bonogi
game 16: start level 32 (ranked #221); win against Obi Wan Bonogi
game 17: start level 33 (ranked #178); win against Obi Wan Bonogi
game 18: start level 35 (ranked #140); win against -Stef-
game 19: start level 36 (ranked #107); win against Obi Wan Bonogi
game 20: start level 37 (ranked #84); win against Obi Wan Bonogi
game 21: start level 38 (ranked #68); win against -Stef-
game 22: start level 39 (ranked #51); win against Obi Wan Bonogi
game 23: start level 40 (ranked #38); win against Marin
game 24: start level 41 (ranked #29); win against Obi Wan Bonogi
game 25: start level 42 (ranked #25); win against -Stef-
game 26: start level 43 (ranked #21); win against Marin
game 27: start level 44 (ranked #19); win against Obi Wan Bonogi
game 28: start level 45 (ranked #14); win against -Stef-
game 29: start level 46 (ranked #10); win against Obi Wan Bonogi
game 30: start level 47 (ranked #5); win against Marin
game 31: start level 47 (ranked #4); win against Obi Wan Bonogi
game 32: start level 48 (ranked #1); win against -Stef-
game 33: start level 49 (ranked #1); win against Marin
game 34: start level 50 (ranked #1); win against Obi Wan Bonogi
game 35: start level 50 (ranked #1); win against Obi Wan Bonogi
ranked #1, level 51
I don't know if this is the fastest way to do it. (Like, is it ever better to win against a lower-ranked player, maybe because they have a smaller variance?)