But I would recommend to do the walk up there instead, otherwise it would not be much of climbing left to do the summit.
Once up on the Eagle nest you continue on the Mt Kosciuszko summit walk. About 6,5 km each way. It is a built up steel walkway so very easy walk and no orientering skill needed:) It's just to stay on the track.
I started the walk late so only met people on their way back from the summit, some school classes but except from that it were not many walkers.
Walk down is easier, if you back at the lifts before 4.30 pm you can save 600 m of descent. Otherwise just walk back same way. Got back in Thredbo at 17.50 and could feel it in my legs then that I had done around 20 km and 1000 vertical meters up and down.
A sun hat can be good for sunny days, wind jacket/rainjacket, sun glasses, some good walking shoes, drinking water and some snack. Don't forget to cut the toe nails and tie the shoe laces tight, could save your from some pain the next days:)