"I have his number."
Translation:Ik heb zijn nummer.
Yes, that would be valid (but not the literal translation!). You are right that 'zijn' is probably a better alternative here, though.
Can " Ik heb zijn nummer " mean " I have his phone number" like in English?
That would be the object pronoun. You have to use the possessive pronoun zijn instead. Hem would be correct in this example:
Ik heb het nummer van hem.
which would literally translate to
I have the number of him.