The only thing you can do to heal from Stun drain is rest, though a Stim patch would suppress it temporarily. If the drain is physical, then you can use first aid or magic to heal it.
Not exactly. SR5 p.278: "Drain damage, regardless of whether it is Stun or Physical damage, cannot be healed by any means other than the natural properties of the body -- that means no magical healing and no medikits."
[EDIT] Regarding the OP, a Medicine + Logic [Mental] Test adds bonus dice to a character's own natural recovery dice pools, so yes, I think Medicine could help recovery from Drain. No bonus from a medkit, though, as that's specifically disallowed.
Just keep in mind that the Medicine test adds to the healing test,
after the allotted time is up for the healing test. So while Medicine can greatly speed up the amount of time needed overall, it can not remove damage boxes (in this case drain, either physical or stun) "right away"...
So, after an hour of rest, a mage could attempt a Body + Willpower + (Medicine hits) extended healing test to remove stun damage from drain.
After a day of rest a mage could attempt a Body x 2 +(Medicine Hits) extended healing test to remove physical damage drain.
Drain sucks. Avoid it, manage your spell casting!