Your chances are pretty good for just showing-up and finding something eventually, assuming there's not some big Expo, concert, or other event going on at that time. You may have to buzz a number of ringers before finding one, however. Hopefully they can tell you if they have an available double before climbing the one, two, or three flights of stairs with your luggage. Many of these places don't have elevators.
It would be smart to take a look at the
websites for and have a list of the names of half-dozen possible places along/near Las Ramblas, just to get an idea of how they look and the prices they're asking.
Enjoy Barcelona! I know you will.
Saludos,
BarcelonaMan