The main character wakes up in a rock bed from getting revived. I think it was snowing.
From this moment he meets his servant that waited for his reincarnation, the story progresses and she admits him into an academy.
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You are looking for Rettou Me no Tensei Majutsushi ~ Shiitage Rareta Moto Yuusha wa Mirai no Sekai o Yoyuu de Ikinuku (or in English THE REINCARNATION MAGICIAN OF THE INFERIOR EYES)
Abel was a genius magician who held immense power in a world where the color of your eyes indicates how powerful you are. However, the strength that came with his amber eyes was so great, even his comrades looked at him with fear and contempt. He, therefore, decided to transfer his soul far into the future, hoping to find a paradise that accepts him. He would soon find out that people’s attitude regarding his eye color did change, but not quite the way he had imagined…
The main character wakes up in a rock bed from getting revived.
I think it was snowing.
As you said, the MC will later go to the academy of magic.