Updating the MST of a graph G = (V,E) when decreasing the weight of one of the edges that is not part of the MST
You are definitely on the right track. As D.W. notes, you have to prove two things:


The new tree is a spanning tree.
There is no spanning tree of lower total weight (i.e. minimality).


For (1), consider what it means to be a spanning tree:


Every vertex must be a part of it (this is the spanning part).
It must be a tree (i.e. a connected acyclic graph).


So with your construction, you have a starting MST $T$ for the graph $G$, and you create a new potential MST $T'$ for $G'$, which is $G$ with a single modified edge weight (which may or may not be the added edge).


Is every vertex of $G$ (and hence $G'$) in $T'$?
Is $T'$ connected?


For (2), you want to show that there's no other tree that is better. Proofs of this kind are normally done by contradiction. So start by assume there is a tree $R$ that is a spanning tree of $G'$ and has lower weight than $T'$, what would this imply?

In thinking about this consider what it means for $R$ to be different, and what this would imply for $T$, the original MST.