Plugging up ceiling vent hole...using backer wood to attach drywall
Using backer strips to hold on drywall patches is a standard repair technique. Often the screws holding the backer boards are inserted through sound plaster or plasterboard from the finished room side.

Your approach seems fairly sound, but I would add a few screws from the good side, being careful not to crack the plaster too much as you install. Since you seem to have real plaster, I would predrill the holes, although when screwing through drywall, it is not necessary.

Finally, are you making some arrangement for a real vent to the outside? Moisture vented to an attic space is simply wrong, leading to possible moisture damage and mold. Closing off the bad vent is a start, but not the full solution for the problem. As @JamesOlson points out in his comment, an openable window is adequate ventilation; but if there is none, some other exterior vent is needed.