What you're looking for is the definition of the Casting Time entry in a spell's description, on page 129 of the Player's Handbook. (Segments are from AD&D 1st edition and aren't relevant if you're playing AD&D 2nd edition.) The relevant part on that page is:
If only a number is given, the casting time is added to the caster's initiative die rolls.
The caster rolls initiative, most certainly. (This is assuming you're using individual initiative, since it sounds like it, but that only affects the details, not the basic answer.) The casting time is how much slower the spell makes the caster, because once they start acting, they still need to start and complete the casting process. (It's worth noting that if they are hit between their raw initiative count and their initiative count + Casting Time, the spell is lost.)
Magic missile's spell description lists Casting Time: 1 (p. 136). Assuming for simplicity that no other initiative modifiers are involved, that means your friendly neighbourhood mage needs to roll 1d10 for initiative (per the Standard Initiative Procedure, p. 94) and then add 1 to find out when the magic missile casting completes. If the mage flubs the roll and the d10 comes up 10, that means the mage casting magic missile isn't acting until initiative count 11. That's likely last or close to it.
Magic missile is faster than most spells, but it doesn't overcome a mage whose reflexes are slow.
Even in 1st edition, initiative affects “beginning a spell” (1st edition PHB, p. 104). If that's the source of his confusion, he's misread or misremembered the rules, or perhaps mistaken an old house rule for an actual rule.