Hi, I am walking in June and from what I Gather it could be cold at night and in the Morning. 4 years ago I walked from Astorga to SDC in June and I had all sorts of weather, from mid 30's to low teens, with Rain.
I did the Portuguese route early Sept and was unlucky to have a week of rain, and the week prior to my walk there had been a heatwave.
Galicia/Austurias are pretty green for a reason, they get rain.
As regards getting to Oviedo, I booked with ALSA on their luxury buses from Santander to Oviedo, it was 13 euros and its cheaper when you book in advance.