I leveled PUP up to 99 on a mule almost entirely solo with a few Trust NPCs and Soulsoother (WHM) Automaton. Your puppet should be able to cast Dia at Lv. 1 with it's base form. Get the RDM Automaton at Lv. 10 and begin using that for magic skill ups as well as some melee skill ups. Stick with that until 40 when you're able to obtain the WHM head which will gain you access to more spells.

Any spell that is cast on you should give it skill ups. That includes Cure, Pro/Shell, and any debuff removing spells like Paralyna, Poisona, Silena, Erase. Also, dependent on level, any debuff that is used should also give skill ups. Dia, Slow (Earth Maneuver), Paralyze (Ice Maneuver) all work.

It won't all happen right away, or in the span of a few hours. But a great way to skill up your mage Automaton while making a little gil for yourself is to do Dynamis content. Nearly all mobs in Dynamis zones will give you skill ups all the way to cap.