It definitely works; I repair my damaged ASFs regularly. Be aware that it takes close to forever to repair internal systems such as Sensors, Targeting, Landing Gear, and Life Support, as in around 10+ full days at the base repair rate, and a lot longer if using extra time. Even armor repairs take something like 3x or 4x the amount of time as it would for a 'Mech or vehicle.
....and it does require an AeroTech, not a MechTech or Mechanic. Note that dropships, jumpships, and warships require CREW who can be assigned to repair their own ship.