Which facilities (e.g. cable cars, museum, viewing areas, etc.) are within this exclusion zone? Does the prohibition mean that the top of the mountain is entirely inaccessible, or just that areas closest to the crater are off-limits?
There is further information (in Japanese) about the different levels on this page from the Japan Meterological Agency. The first image (with the concentric circles showing different distances) also has a bit more information about what facilities you don't have access to.
Level 2: 1km exclusion zone around crater. No access to Mt. Aso Park Road (阿蘇山公園道路), Mt Aso Ropeway (阿蘇山ロープウェイ）, Sensuikyo Ropeway (仙酔峡ロープウェイ). If you look at this wider scale map, you'll see that the closure of Mt. Aso Park Road basically removes access to all the near-crater facilities. I think the orange line on that map marks the current restriction.
There is more information about the closure of Mt. Aso ropeway on their official site (English), and this would probably be the best source for seeing if things are open again as they seem to update regularly to let people know if they will be running.
It appears that there has been damage on some of the roads/pathways in the area from earthquakes, including ones linked to Mt. Aso ropeway, so even if the volcanic alert level drops, you should double-check whether the areas you're interested in are both open and accessible - this is especially true if you're thinking of hiking in the area.