I think we still need more information about your server and requirements, but the most straight forward way of updating a file on a web server is FTP.
Point the FTP server to folder that is used by the web server to serve the image. Most hosting providers will offer some sort of FTP solution.
On your client machine rename your local file to be the same name as the server file name then use an FTP program (can be command line or GUI) to upload and overwrite the server copy.
The one thing to watch out for this solution is caching. If a browser has already viewed(downloaded) your image then visiting it again after it has changed may show them the old copy from the browsers cache. There are lots of threads on how to "cache bust" and "cache busters" to avoid this issue.