"The year just started" is past tense. "The year just begins" is present tense.
"My brain in fact was only just beginning to wake up." How would one say it in italian?
"Il cervello, infatti, appena cominciava di svegliarsi"
You don't say "il mio cervello" if it is easy to assume who's body part you are referring to. You say "cominciava" as it is the imperfetto form of cominciare - "to start". It is the equivalent of the English past continuous tense.
Svegliarsi is a reflexive verb.