(Disclaimer: assuming we can actually see the mountain during our trip…
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The bus to take us from Kurokawa Onsen to JR Aso Station is the Cross-Kyushu service from Beppu to Kumamoto. Since we will not be able to go near the crater, instead of going to Kusasenri and returning to JR Aso Station, we can continue down the eastern side of Mt Aso (or strictly speaking Nakadake) to Tateno JR Station.
Japan Guide
Google Map showing the size of Aso Caldera with Kurokawa-sō near the top, Aso town, active volcano (red drop-pin), Tateno and Hakusui Kōgen stations (yellow stars) lower left
This will mean we can only see Mt Aso from the bus (assuming we can see it at all!) but one of the best views of the volcano appears to be on the way from Kurokawa Onsen to Aso Station.
By Miya.m via Wikimedia Commons
Tateno seems to be a place with nothing more than some houses and a railway station (actually two stations). One reason why Limited Express trains from Beppu to Kumamoto stop at Tateno is because the station is part of the zig zag system of JR Hohi Line.
National Land Image Information (Color Aerial Photograph), Ministry of Land, Infrastructure and Transport
Brown lines show JR Hohi Line with Tateno Station on the right and Kumamoto to the left
Tateno is also a terminus of Minamiaso Railway which used to be part of the JR system. Our bus should arrive in Tateno at 12:09 and the next Limited Express train to Kumamoto leaves at 14:10 (arriving in Kumamoto at 14:49). According to Google Street View there is a noodle restaurant on the main road where the bus will drop us. Although there is an earlier local train to Kumamoto, I'd like to suggest the following detour.
There is a Minamiaso Railway train leaving Tateno at 12:46. We can travel 15 mins on this train to the ridiculously named 'Minamiaso Mizu-no-Umareru-Sato Hakusui-Kōgen Station' (Hakusui Plateau, birthplace of Minamiaso mineral water). We will wait 7 minutes for a train travelling in the other direction back to Tateno. We should be back in Tateno at 13:21.
The bus fare from Kurokawa Onsen to Tateno will be 1,340 yen instead of 2,130 yen for Kurokawa Onsen-Aso Station-Kusasenri-Aso Station. The Minamiaso Railway is not covered by the Northern Kyushu Rail Pass but if the weather is kind to us it should be a scenic ride.