Determine if a vertex is a part of a cycle in O(m+n) complexity
Simply apply Strongly connected components algorithm , Kosaraju's algorithm has complexity $O(m+n)$. Now, read through the SCC components and check if $v$ is part of a component with size $\geq$ 2. The second step is $O(n)$. Actually, the second step can be interleaved with the first step to get your required complexity result.

A simpler algorithm may be just to to use DFS on the graph. Have a look at Cormen's Introduction to algorithms, chapter 22. Your exact problem is solved there using DFS.