1-Can one say "the night before" instead of "last night".
2-Can one say "the time before" instead of "last time".
One of the most famous poems in the English language begins, "'Twas the night before Christmas...." It does not there and it is unlikely that it is ever used to mean "last night". (Ditto TDOL's other comments too.)
Here are possible usages for "the night before":
He was hung over Thursday morning. He had stayed out all night the night before.
That happened the night before last.
In what context were you considering using those expressions?