Pain will / shall pass, but pride will last forever.
Your friend is right; "shall" would be quite wrong.
Traditionally, to express futurity, "shall" is used for 1st person subjects and "will" for 2nd/3rd person subjects. The reverse applies where the meaning is one of volition.
In your example, "pain" is a 3rd person subject and since the verb expresses futurity, not volition, it really should be "will".