@xbirchy... without wanting to offend or sound condescendent... I believe you're very unaware of the distances and timings.
Distances: Barcelona isn't particularly big if compared to other major capitals of the world, but the city is by no means "small"... we're talking 40 sq miles. While many visitors won't necessarily visit certain areas of the city, note that the "typical" places you're bound to visit are not in one area but spread across several districts of the city. There's a great public transportation network (bus,metro,train,tramway... and taxis, of course) so, despite being one of the most condensed cities in Europe (some districts have a density of 41000 inhabitans/sq mi.), it's fairly easy and relatively fast to move about... but still...
Timings: aside of the reflection made above regarding lines at certain places (even if you get the ticket in advance), another reflection to take into account is what are your expectations when you're visiting "a site"... is it just to tick a list? or are you the type to enjoy the details and the stories of the site you're visiting?... Unfortunately some cities, and Barcelona is one of those, have too many sites worthwhile visiting for different reasons: architecture, art, history, entertainment, shopping, etc, etc. so it's always a nightmare to "choose". Some visitors opt for seeing "as much as possible"... your choice, of course, but sadly you're not really going to soak much.... My advise is always to limit your itinerary knowing that you're going to be leaving a lot out, this way, if you focus on a limited number of sites/places, you'll be able to really get to know a bit of the city. Just sayin'
Some notes:
Montjuïc is a whole district with over two dozen things to do/see spread over a large area: http://www.barcelonaturisme.com/wv3/en/page/396/parc-de-montjuic.html Btw, the name "park" --a marketing gimmick that only the Tourist Information Office uses!... don't use it if asking a resident cause no one calls it that, we use simply "Montjuïc" (mon-jew-eek)-- might cause confusion... neither it's an enclosed commercial premises (ie like an amusement park) nor it's a park in the flora and fauna sense. In truth, no one really understands why the TI has recently added the word "park"... other than to create a marketing package... regardless the reason, it's not a good idea because it complicates defining an itinerary when you don't know the sizes and distances :))
El Gòtic (Gothic Quarter) is one of the six neighbourhoods that form the old city (Ciutat Vella), again, it's a whole neighbourhood so not something you "see in a couple of hours" nor it has a "marked itinerary" --Ikea style-- to see it with a pre-determined entry and exit points. This 2 sq mi. area is full of sites and things to see as it's the oldest part of the city, founded in the year 215 BC, from Roman ruins to Medieval palaces, lots of old narrow streets, thousands of small shops and restaurants, several churches, markets, boulevards.... One could spend a whole day just strolling this area... without even visiting any specific site.
I don't want to seem difficult, lol!, nor to spoil anyone's plans.... I am just trying to convey what you're going to encounter in-the-field when you're here.
As a rule of thumb -because it depends on which sites one chooses and where are they in relation to each other- I recommend to plan not more than three sites per day noting that 99% of the visitors do also stroll and wander the streets.