Aso gets all of 15% personal approval in a Mainichi News poll which probably translates to the low 20's in non-partisan terms (Mainichi is not a great fan of the LDP). In a similar Kyodo News poll, Aso gets about 20%. Support for the Aso government is stated to be 19% and 17% respectively - call it 20%:2. Even the LDP-friendly Yomiuri poll (which didn't ask for, or doesn't disclose, Aso approval figures) found only a bit over 20% support for the cabinet. Not a level most politicians would be comfortable with, less than three months before an election.
It is worth pointing out that Hatoyama (the elder brother, the new DPJ leader) gets about 30% support in the Mainichi poll, which is much better than Aso's figure but he is still beat by Mr. None Of The Above at around 45%. The Yomiuri poll seems to have asked people to specifically pick between Hatoyama and Aso and found Hatoyama lead with 45% over 25%; no word on what the remaining answers where Hatoyama got over 50% in the Kyodo poll, which leads me to suspect their question was similar to Yomiuri's, and perennial favourite None Of The Above was not an option3. The DPJ gets about 40% versus LDP's 20% in the Kyodo poll, 35% and 20% in the Mainichi poll, and 30% versus 25% in Yomiuri. Again, the party of None Of The Above seems to be at least as popular as the DPJ.
In other words, DPJ is not tearing up the boards here. It's popularity rests simply on the virtue of not being the LDP. If it manages to win the next election, I expect a brief and unenthusiastic honeymoon with the electorate ending in the inevitable Narita Rikon4 once the voters realize their new partner is no different from their previous one.
#1 A different Hatoyama - yes, they're brothers, and yes, I'm happy there isn't a third Hatoyama brother around in Japanese politics; it's all too similar to a Korean soap-opera already.
#2 Giving two- or three-digit precision for polls like these is pointless and actively misleading. Polls of this magnitude, and especially with a skewed, partially self-selected sample (only about 2/3 answered the questions), rarely have better precision than 4-5 percentage points or so.
#3 This is the main problem with these kind of poll articles: they don't give you the methodology, the error estimates (which vary from question to question) or the actual questions asked. It makes them impossible to compare with each other, or even to estimate the true change over time for the same poll.
#4 Narita Rikon - Narita Divorce - is when a couple gets married without first getting to know each other, discover on their honeymoon that they actually dislike each other, and immediately file for divorce on return to Narita airport.