When will be the best time of the day to visit Mount Aso? Morning or late afternoon?I don't think there is a difference.
If I reach the Crater using the last bus from JR Aso Station (15:20), will I have sufficient time to see the Crater and it's surrounding (Kusasenri) in order to catch the last bus (17:00) from the crater to Aso station?
Yes, that should be enough time to take the ropeway and see the crater. Note that sunset can be as early as around 5pm during winter. On overcast days it could be already quite dark, but you should still be able to see enough.
Additionally, there is always the risk that you cannot go to the crater due to volcanic activity. Current conditions can be viewed here:
http://www.aso.ne.jp/~volcano/eng/index.html