How can this grammar parse such an input?
In an attribute grammar one assigns values to the nonterminals of the grammar. This is to help evaluating parts of a grammar that are not context-free. One of the simple types is inherited attributes, these are assigned bottom-up in the derivation tree.

The indexes $S_0$ and $S_1$ are used here to distingish different occurrences of the nonterminal $S$ in the grammar. A good choice for an attribute here would be the difference of the number of occurrences $b$ and $a$. This number should never be negative according to your specification. I do not know how you are supposed to denote this, but assume the attribute is called $N$. For the rule $S_0 \to S_1 a$, the value of $N$ for $S_1$ determines the value of $N$ for $S_0$: $S_0.N = S_1.N - 1$ (since $S_0$ has one more $a$ than $S_1$.

Similarly for the other rule that introduces a $b$.

Now we have to introduce another attribute. In the end we should test whether the last symbol derived is a $b$. So the attribute should remember that letter and test it in the last step $V\to S$ (this is the last step in a bottom-upevaluation).

Again: check your class notes on how to specify alle these details. 

Your examples are not clear: $aa$ does not end in a $b$ and neither has it at least as many $b$'s (?!)