Learn English – It is correct to say “on level” or “at level”

Level can mean two things - how something measures against a scale, and a platform one stands upon.

If by "level" you mean "how high, how much," at is used. Think of a scale, like that on the side of a measuring cup, and with this use of level you are asking where the thing you are measuring lies on such a scale. You would typically use the preposition at when answering where.

If by "level" you mean a platform one stands upon, on is used.

Also, @MrTheWalrus is 100% correct, on a certain level is an idiom and should be treated as a special case. So is on the/that/a same level:

Her and I were on that same level.

This is an informal (if not slang) idiom that means the two people here had a mutual understanding about something, likely not explicitly spoken.

You could consider "the next day" as a adverbial phrase, so the sentence

He came the next day

is grammatical and idiomatic.

Another example from Collins dictionary could verify this kind of use.

But the next day my solicitor contacted me to say the money had not arrived.