"No Spells will affect you"
"Spells cannot affect you"

so...

Is this sentence related to the target of the Spell? Or is it just RAW?

so...

Can the Barbarian (or a character with Amulet) pass the Barrier Spell without stop his movement?

Other similar questions:
Is he affected by other Spells targeting spaces, like Hex Spell or Black Ice?

The Barbarian or a character with an Amulet are not affected by any of the spells you mentioned.