Finding a '1' cell with a '0' to its right in a binary array
After viewing the original question, it seems that you have an additional piece of information: you know that the first element is 1 and that the last element is 0.
This is crucial!

So indeed you can solve it with binary search: first observe that if the first element is 1 and the last is 0, there must be subsequence of the form $1,0$ in the array. 

Denote the array by $A[1,...,n]$. If $A[n/2]=1$, then there is a $1,0$-subsequence in the subarray $A[n/2,...,n]$. If $A[n/2]=0$, then there is a $1,0$-subsequence in the subarray $A[1,...,n/2]$.

So you can use Binary search, which takes $O(\log n)$ time.