Greasing A Manual Gearbox

If seals are good and you can drain out the old , just refill with 80w90.

In my repair world when we work on a rotary mower gear box that is subjected to a lot of crap wrapping around the shafts and tearing the seals, the oil does not last at all. We do just that Rusty. We change the seals and put the grease gun in the hole (typically 3/8” or 1/2” NPT) and pump till it’s full. Leave a little room for expansion and cap.
Way more heat and revolutions than a steering box. I’m sure you will be fine.
Easier to heat the grease up till flowable and pour it in with a funnel I bet. Just don’t burn the grease.