What is the time complexity of finding the residual network in this case?
The time complexity of this problem is (at most) the time complexity of the network flow problem.  You are given a graph and you want to find a flow of size $f$.  You can use any network flow algorithm for this.  Different algorithms have different running time.  Wikipedia has a list of algorithms, with their running times.  There is not a single optimal algorithm; different algorithms work better in different situations.

You mentioned you want to find a flow of size at least $f$, rather than a max flow.  Many of the algorithms for max flow can be adjusted to stop early when you've found a flow that is large enough.  Or, you can adjust the graph to ensure that a flow of size $f$ is a max flow -- if the original source vertex is $s$, add a new vertex $s_0$ and an edge $s_0 \to s$ of capacity $f$, then treat $s_0$ as the new source vertex and solve the max flow problem.

Once you find the flow itself, you can compute the residual network in $O(m)$ time.  For each edge, you compute how much remaining capacity it has (the capacity of that edge minus the amount sent by the flow); and you add a reverse edge if necessary (the amount of the flow).  This is $O(1)$ work per edge.  All known network flow algorithms take $\Omega(m)$ time.  Therefore, the time to compute the residual network once you find the flow is negligible; the bottleneck is finding the flow itself.