I've realised that the rank that a player has is not updated once a game has finished (I suppose that it's updated once a day) but, how is possible that a player with a higher rating had a rank position lower than another with a lower rating? Perhaps the rank position is not updated simultaneously for all players?

Excuse me if you already know this,and you probably do, but the lower the  numbered rank (#4 is better than #5) the better.  hope that helps.

Yes, I do. I'll illustrate with an example

1596 is better than 1590. Despite of this the ranking is lower (the number of the position is higher, of course). The information of these screenshots appears right now on the ranking information. 

I'm going to take a guess and say that it's because rank is only updated once a day, while rating is updated after every game. At the start of the day the 1st player had  a lower rating than the 2nd one, but won a game, or the 2nd player lost a game, and the ratings where updated. The ranks will not reflect the updated rating until the next day.

Yes, it's probably that the reason for this contradictory information could be what you say. Perhaps rankings should update as ratings do? Or, perhaps that's not possible due to some software limitations? Despite of all.. congratulations for this great site!!

Updating a rating isn't that intensive, but updating a ranking would probably really slow down the site.