Is it correct to use the preposition in this case? e.g.On the next day,he really felt better.
Welcome to Using English.
Don't use the "on" there. Just write "the next day."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
But it would be OK (?) if it didn't start a sentence. "On Friday he started to feel ill and went home early. He spent Saturday in bed. On Sunday he was beginning to feel better; and on the next day he really felt better." (I'm not sure about this... I'm beginning to think, in that context, I'd say 'and on the Monday...'... Hmm... Stick with Barb's answer
)
b
Not "on." You could say "by the next day, he felt better."