Should work okay. Big syringe with a large tip - maybe a short length of tubing on the tip.Going to try to suck it out with a syringe. Hope it works. Don't want a big mess to clean up.
You're the one responsible for me buying that MightyVaci use a fluid evacuator for that too, among many, many other uses. refill with a hose on the end of the gear oil bottle's nozzle.
Probably does not really matter here but learning from car gear boxes and differentials I always open the fill plug first. Reason being is if you cannot open the fill plug you still have a driveable vehicle and can go to a work shop for help whereas drain plug opened and no oil in the gear box you would have a problem. With the mower you can still cut the grass and deal with the seized plug later.If i am reading this right, remove both plugs to drain, then put lower one back in and squirt oil in top hole until it starts coming out/the amount it calls for.