It sounds like you have an exhaust fan already, and plan to cut a hole in both sides of the wall between the closet and main room, then add one of those wall plates on the inside of the closet.
You'll want to somehow seal the gap between the interior of the wall so you don't get humidity and mold in between the drywall. I'm not sure that stub end is long enough to pass through both sides of the drywall. You could add one of those wall plates to both sides of the wall, put a length of
galvanized duct pipe between the two, and seal with
silicone.
When you add an exhaust fan and ducting to that line, the gap between your closet door should provide some intake flow. If you need more, it's not uncommon to cut a hole in interior pre-hung doors to increase air flow and just cover with a
grille on each side (
example here).
Inside the closet, you'll want to run ducting from the wall plate to the far side of the closet away from the door (and up high, if possible) to promote mixing.
Outside the closet, you'll still want to run ducting away from the closet door intake to avoid recirculation. If you can get away with it in your area (e.g., neighbors, etc), a
window dryer vent kit is probably best if you are not going to do any dedicated external vents.
In general, fans work better by pulling air rather than pushing air - which is why you typically rely on exhaust fans and passive intakes instead of vice versa.
