I walked the Primitivo in August 2017 (also first Camino). I took the train from Barcelona to Oviedo (change in Madrid). Long journey, but I'm not much for flying when I can avoid it, so it was the right choice for me.
I reserved almost every night. A day in advance should be fine most of the time. I remember that early on, there were a lot of people very much against reserving anything in advance, ever. By the end, pretty much everyone I met did, as everyone was tired of having to walk on or call a taxi to the next town. There just aren't as many albergues on the Primitivo as there are on the Frances, and people end up walking the same distances and stopping in the same villages. There wasn't a night when the albergues I stayed at weren't completely full or with just 1-2 bunks left at most. And when the next one is 15km on or more, things gets tough. I remember Berducedo (after the highly!!! recommended Hospitales route) being particularly competitive, and that village had at least three albergues, all full.
The one I would recommend reserving more than a day in advance is Bodenaya, should you plan on staying there, as only a few beds are up for reservation, the rest is first come first served. And that albergue is absolutely magnificent.
I took the cheapest sleeping bag available, no problem at all. Most albergues even had blankets (sometimes for a little extra charge). It can rain a lot during the day, but it didn't really get that cold during the night.