python en 44 LoC, 110ms et 10MB RAM

import sys
I = sys.stdin.read().splitlines()
G = [list(l) for l in I]
R = len(G)
C = len(G[0])
def rotate(G, F):
 for r in range(R):
 for c in range(C):
 F[c][R-1-r] = G[r][c]
def tilt(G):
 for x in range(C):
 for y in range(R):
 if G[y][x] == "O":
 for j in range(y-1,0-1,-1):
 if G[j][x] == '.':
 G[j][x] = "O"
 G[j+1][x] = "."
 else:
 break
score = lambda: sum((R-y)*row.count('O') for y,row in enumerate(G))
cache = {}
F = [['?' for _ in range(C)] for _ in range(R)] # pre-allocate saves 10% time (no GC)
end = 1_000_000_000
i = 0
while i<end:
 i += 1
 for j in range(4):
 tilt(G)
 if i == 1 and j == 0: print(score()) # part 1
 rotate(G, F); F, G = G, F
 h = ''.join(''.join(row) for row in G)
 if h in cache:
 period = i-cache[h]
 skip = (end-i) // period
 i += skip * period
 cache[h] = i
print(score())