I see more encouraging weather averages for late November, though it's true that historical stats do not guarantee the future. Looking at the average-temperature table on the city's Wikipedia page, I'd expect the average high to be in the low 60s and the average low to be in the low 50s. It could well be down in the 40s on some nights. It's a bit worrisome that all those Wikipedia weather charts seem to cut off in 2010, and weather seems to be a lot more volatile now.
Montserrat is likely to be chillier than the city (maybe windier??), and it takes quite some time to get there. It's a trade-off between doing that and spending the time seeing places in the city that are easier to get to.
However, many of the top city sights pretty much require advance purchase of tickets, because you do not have time to stand in line. That applies to La Sagrada Familia, Parc Guell, the Picasso Museum, Casa Batllo, Casa Mila and the Palau de la Musica Catalana. I am not saying those would be my top picks, much less that you have time to see all of them, just that you need to investigate the ticketing situation if any of them interest you.