A solution based on static variables may look attractive, because it is easy to implement. However, it quickly becomes a maintenance liability, particularly in a web application environment.
The problem with static variables in web environments, such as IIS, is that they are not shared globally across your entire application: if you configure your app pool to have several worker processes, each process would have its own copy of all static variables. Moreover, if you configure your infrastructure for load balancing, each process on each server would have its own copy, with no control on the part of your application. In your situation this would mean a possibility of multiple updates happening at the same time.
That is why I would avoid using a static variable in situations when it is absolutely critical that at most a single request be in progress at any given time.
In your situation, the persistence layer should be in charge of not corrupting the data no matter how many updates are firing at the same time. Persistence layer needs to decide which requests to execute, and which to throw away. One approach to solving this problem is optimistic locking. See this Q&A for general information on how it could be implemented.