Our train arrives in Barcelona from Madrid at 12:30. We are staying in Old City. Can we realistically get to and settle into hotel, metro to the Sagrada Famillia, do the audio tour at 3pm, and towers, then taxi up to Park Guell before it closes on a Friday night? Is Friday busier than other days in July? Thanks!
From the time you arrive at the train station (Sants most likely) and get to the "old city" and in the hotel, it will be one hour minimum, and most likely 90 minutes....then to Sagrada Familia...you will have timed admission to the towers...by the time you get to Park Guell, it will be late and you will be tired. I would stop with the Sagrada Familia and take time to enjoy it more than just running through...
Park Guell is open until 21:00. Advance timed tickets can be purchased and entry is allowed up to 30 minutes after your time. If you purchased 19:00 tickets that would give you almost 4 hours at SF and 2 hours at Park Guell - should be plenty of time for both. I imagine both places are at capacity in July so embrace the crowds and enjoy!
I think you could do it, but why?
Sagrada is a place where the definition "unique" truly applies. Allow time for you mind to become fully engaged. I recommend you go up at least one of the towers.
Afterwards, take a stroll up avenue Gaudi and stop to enjoy a couple of tapas and drinks. Too bad you will not be there at dusk for this street becomes alive with local families taking a stroll offering a glimpse of local life. However, you can walk up to the hospital and catch a cab from there to the Parc.
Unfortunately the Parc has become a crush of tourists now enclosed by a fence and believing that since they had to pay to enter they must max the time in order to get the most for their money.
It really boils down to how long you want to spend in SF and in PG. For me - after about 4 hrs in most places my brain is on overload anyway.