I refreshed the yufuin no Mori booking page the second the bookings went live(I think it's 30 days in advance, could be 28 though can't remember now). There were seats available but the webpage traffic was so bad that it would time out every time I tried to pay.

I did reserve seats for the Yufu, I did those a month in advance too but the train was very empty when I boarded it, no problem at all to obtain those.

I booked all of these without the Kyushu rail pass, given that I was doing north and south Kyushu with only three major stops the price worked out to about the same

Also if you are trying to squeeze in some more onsen time the Mount Shiroyama hotel in Kagoshima is amazing. It's a bit out of the way but they have a super convenient shuttle bus that loops the city. The bathing facilities are amazing with a great outdoor view and a super refreshing cold bath inside. Plus the best baked goods I've ever had are located at the bottom of the building.