How to change the algorithm implication by changing the invariant?
Well, the new invariant does not hold in the code you posted, so I guess you are asked to change the code accordingly.


  What I firstly supposed is that the summation must be done in reverse order.


That might be a wise choice. You could try something like this:

$$
\begin{array}{l}
sum,i:=0,n;
\\
\{\text{Invariant}\ P: 0\leqslant i \leqslant n\ \wedge sum=\sum_{j=i}^{n-1} b[j]\}
\\
\{T:i\}
\\
do\ i\neq 0 \rightarrow\ sum, i:=sum+b[i-1],i-1;\ od
\\
\{R: sum=\sum_{j=0}^{n-1} b[j]\}
\end{array}
$$

I have not checked that this Hoare triple is valid, but it should be, possibly with some minor changes.