I'm making my GAE app using Eclipse. All the plugins, and other functionalities involved support development for the Standard Environment. Is there a way to convert my current project to the Flexible Environment? Barring that, once uploaded to GAE, can this change be done online?
The Flexible environment is an almost entirely different beast than the Standard environment.
Support for various pieces of infra differs. You need to check each piece of infra you use in the Migrating Services from the Standard Environment to the Flexible Environment guide for an equivalent, if there is one. Effectively you'd be re-writing the app.
Note the service in above article title - the environment choice is done at the app service/module level, one can mix services with both environments inside the same app.
And no - the change can not be done online, you need to explicitly deploy the re-written app service to switch its environment.