Mr.Cruiser

Yeah, you just get the 'extended' in-person version of the 'new' muster drill.  I assume it's similar to what happens if you don't do it on the app.
It didn't take that much longer.  We were directed to a nearby lounge one we boarded, and they went over the usual things, and then we were handed over to another crew member who took us to our 'real' muster station and let us go.
After that we went back to the royal promenade and found someone to help us book our shows. 

We took the port shuttle bus that dropped us off at the cruise terminal that is closest to the Columbus monument, and then just walked around.  The Virgin Voyages ship was docked there that day.

Luckily, I had no issues with photos on our cruise; was able to find all of them both pre and post Barcelona. 
Also had no issues with dining.  We had traditional dining; seemed like our entire section were people that embarked in Rome.  We didn't try doing any specialty dining on this cruise (other than some onion rings at Playmakers). 
My teenage daughter was also bummed about new friends leaving halfway thought the cruise. 

We just got off the ship and walked into Barcelona - most of it is easily walkable.  We did get on one of the on/off buses in the afternoon which was just OK (and overpriced).
Viator, which @ChessE4mentioned above is always an option if you want to see excursions.  we used them in Rome for private tours during our pre cruise stay.

So sorry this is happening to you and the other 1200 guests.
I had never heard of Royal doing a sailings like this (Thanks @Mr, Cruiser for the link).  Where x number of passengers board on a Saturday for seven days in one port.  Then at then at one of the ports of call they board more passengers who don't get off in the next few days with the original passengers. (I know some cruise line does this with 14 day cruises out of Barbados, where you can board for 7 or 14 days but each one in Barbados)
I can see why Royals IT had trouble with this as it would be very complex programing. The concerning part would be the reservation disappearing altogether, but understandable if they did a cruise reset on the app at the conclusion of the first itinerary.