DecemberJanuary in Barcelona

We were in Barcelona and neighboring area at exactly the same time that you are planning. It was actually busier than I anticipated, but quite manageable. New Years Eve is quite an event so look for accommodations well in advance for that evening. Some restaurants were closed on New Years Day and I think some of the museums. You still will need to reserve tickets online. At least a week ahead for La Sagrada Familia, and I think it was a couple days in advance for Casa Mila and Park Guell. It also still felt a bit busy at Montserrat. We had wonderful directions, I believe it was a metro to a train station and then a gondola. Someone with better information than I, will respond I'm sure. We got to Montserrat at right about opening. Watch the lines for Black Madonna, they build at times during the day. That said, the line moves fast. There is also an option for a funicular that goes up higher than the monastery for amazing views and hikes. I went back to my pictures and we were wearing light weight long sleeve shirts and pants/leggings, our jackets tied around our waists. Make sure to wear sturdy shoes. I brought a picnic lunch to eat at the top. If you want to do the funicular, make sure you pick the correct ticket package. I believe I purchased the tickets at the train station and there were two different booths, competing. However, I explained what I wanted and I was sent to the other booth so they try to make sure you get what you need.
We actually thought the weather was lovely. Typically sunny, one day was cloudy. Highs were low 60s. We even went to the beach at Barcenoleta. Pictures from there show we were barefoot and walking in the ocean in skirts and jackets. Bear in mind that we're from Minnesota. There was still some outdoor restaurant seating, some with heaters. My preference is always to be cooler over too hot. I would definitely go back to Barcelona at that time of year and for me personally I would probably not do any of Spain except maybe Basque in the summer.
Oh, and I think they celebrate Epiphany, which is January 6th, I believe. Our last evening there was some Epiphany/3 kings celebration on the Plaça de Catalunya. We were in Vienna on actual Epiphany and there were places closed because it is apparently a national holiday. I would be sure to check to see what will be open on both New Year's Day and Epiphany.

I don't think there's a time when Barcelona isn't busy. I've been in early February and all the main attractions were very busy. Lovely weather, I was in shorts and t-shirt but the locals were still in their winter gear.